˝antum dynamics in low-dimensional topological systemsprimero estudiamos la dinámica de dublones,...

�antum dynamics inlow-dimensional topological

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Memoria de la tesis presentada por

Miguel Bello Gamboapara optar al grado de Doctor en Ciencias Físicas

Universidad Autónoma de MadridInstituto de Ciencia de Materiales de Madrid (CSIC)

Directora: Gloria Platero CoelloTutor: Carlos Tejedor de Paz

Madrid, Septiembre 2019arX

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Abstract/Resumen

The discovery of topological matter has revolutionized the �eld of condensed matterphysics giving rise to many interesting phenomena, and fostering the development ofnew quantum technologies. In this thesis we study the quantum dynamics that takeplace in low dimensional topological systems, speci�cally 1D and 2D lattices that areinstances of topological insulators. First, we study the dynamics of doublons, boundstates of two fermions that appear in systems with strong Hubbard-like interactions. Wealso include the e�ect of periodic drivings and investigate how the interplay betweeninteraction and driving produces novel phenomena. Prominent among these are thedisappearance of topological edge states in the SSH-Hubbard model, the sublatticecon�nement of doublons in certain 2D lattices, and the long-range transfer of doublonsbetween the edges of any �nite lattice. Then, we apply our insights about topologi-cal insulators to a rather di�erent setup: quantum emitters coupled to the photonicanalogue of the SSH model. In this setup we compute the dynamics of the emitters,regarding the photonic SSH model as a collective structured bath. We �nd that thetopological nature of the bath re�ects itself in the photon bound states and the e�ectivedipolar interactions between the emitters. Also, the topology of the bath a�ects thesingle-photon scattering properties. Finally, we peek into the possibility of using thesekind of setups for the simulation of spin Hamiltonians and discuss the di�erent groundstates that the system supports./

El descubrimiento de la materia topológica ha revolucionado el campo de la física de lamateria condensada, dando lugar a muchos fenómenos interesantes y fomentando eldesarrollo de nuevas tecnologías cuánticas. En esta tesis estudiamos la dinámica cuánticaque tiene lugar en sistemas topológicos de baja dimensión, concretamente en redes1D y 2D que son aislantes topológicos. Primero estudiamos la dinámica de dublones,estados ligados de dos fermiones que aparecen en sistemas con interacciones fuertes detipo Hubbard. También incluimos el efecto de modulaciones periódicas en el sistema einvestigamos los fenómenos que produce la acción conjunta de estas modulaciones yla interacción entre partículas. Entre ellos, cabe destacar la desaparición de estados deborde topológicos en el modelo SSH-Hubbard, el con�namiento de dublones en unaúnica subred en determinadas redes 2D, y la transferencia de largo alcance de dublonesentre los bordes de cualquier red �nita. Después, aplicamos nuestros conocimientossobre aislantes topológicos a un sistema bastante distinto: emisores cuánticos acopladosa un análogo fotónico del modelo SSH. En este sistema calculamos la dinámica de losemisores, considerando el modelo SSH fotónico como un baño estructurado colectivo.Encontramos que la naturaleza topológica del baño se re�eja en los estados ligados

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fotónicos y en las interacciones dipolares efectivas entre los emisores. Además, latopología del baño afecta a las propiedades de scattering de un fotón. Finalmenteechamos un breve vistazo a la posibilidad de usar este tipo de sistemas para la simulaciónde Hamiltonianos de spin y discutimos los distintos estados fundamentales que el sistemasoporta.

Acknowledgements/Agradecimientos

I would like to thank Gloria Platero, my thesis advisor, for taking me into her researchgroup and providing me with all the necessary means for doing this thesis. I alsothank her for her trust, proximity and constant encouragement. Thanks to Charles E.Cre�eld for helping me take the �rst steps into research, and to Sigmund Kohler forbeing always willing to talk about physics. I thank J. Ignacio Cirac for giving me theopportunity to do a stay for 3 months at the Max Planck Institute of Quantum Opticsin Garching. My time there was very productive and rewarding. I thank all the peoplethere for their warm welcome, specially to Javier and Johannes, with whom I have spentgreat times and done very fun trips. There I also met Geza Giedke, who has alwaysbeen keen to answer my questions, and Alejandro González Tudela, whom I thankfor proposing very interesting problems, and also for his closeness and dedication. Ithank Klaus Richter for inviting me for a short stay at the University of Regensburg,and for the enlightening discussions we held there together with other members ofhis team. I thank all my workmates from the ICMM: Mónica, Fernando, Yue, Jordi,Chema, Álvaro, Beatriz, Jesús, Jose Carlos, Guillem, Sigmund and Tobias for creatingsuch a relaxed working environment, the outings and good moments together. I thankspecially Mónica for encouraging me to keep going; Álvaro, for convincing me to go tomusic concerts I will never forget; and Beatriz for making me laugh so much. I thank allof them for teaching me directly or indirectly many of the things I have learned duringthese four years. Thanks to my uncle Jose Manuel for being a constant inspiration.Thanks to Darío for being there in the good and bad moments. Last, I want to thankmy parents, this thesis is specially dedicated to them.

The works here presented were supported by the Spanish Ministry of Economy andCompetitiveness through grant no. BES-2015-071573. I thank the Institute of MaterialsScience of Madrdid (ICMM-CSIC) for letting me use their facilities and the AutonomousUniversity of Madrid for accepting me in their doctorate program. /

Quiero agradecerle a Gloria Platero, mi directora de tesis, el haberme acogido en su grupode investigación y el haberme dado todos los medios necesarios para hacer esta tesis.También le agradezco su con�anza, cercanía y estímulo constantes. Gracias a Charles E.Cre�eld por ayudarme a dar los primeros pasos en la investigación, y a Sigmund Kohlerpor estar siempre dispuesto a hablar de física. Quiero agradecerle a J. Ignacio Cirac elhaberme brindado la oportunidad de realizar una estancia de 3 meses en el institutoMax Planck de Óptica Cuántica en Garching. Mi tiempo allí fue muy productivo yenriquecedor. A toda la gente de allí le agradezco su calurosa acogida, en especial aJavier y Johannes con quienes pasé muy buenos ratos e hice viajes muy divertidos. Allí

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también conocí a Geza Giedke, que siempre ha estado dispuesto a responder mis dudas,y a Alejandro González Tudela, a quien agradezco el haberme propuesto problemasmuy interesantes, y su cercanía y dedicación. A Klaus Richter le agradezco el habermeinvitado a la Universidad de Regensburg y las interesantes discusiones sobre físicaque allí mantuvimos junto con otros miembros de su grupo. Le agradezco a todos miscompañeros del ICMM: Mónica, Fernando, Yue, Jordi, Chema, Álvaro, Beatriz, Jesús,Jose Carlos, Guillem, Sigmund y Tobias el crear un ambiente de trabajo tan distendidoy las salidas y buenos ratos que hemos pasado juntos. Le agradezco especialmente aMónica el motivarme a seguir adelante; a Álvaro, por convencerme para ir a conciertosde música que nunca olvidaré; y a Beatriz, por hacerme reír tanto. A todos ellos lesagradezco el haberme enseñado directa o indirectamente muchas de las cosas que heaprendido durante estos cuatro años. Gracias a mi tío Jose Manuel por ser una fuenteconstante de inspiración. Gracias a Darío, por estar ahí siempre en los buenos y malosmomentos. Por último, quiro darle las gracias a mis padres, esta tesis está dedicadaespecialmente a ellos.

Los trabajos aquí presentados fueron posibles gracias al apoyo económico del Mi-nisterio de Economía y Competitividad a través de la beca n.º BES-2015-071573. Leagradezco al Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC) el habermepermitido utilizar sus instalaciones y a la Universidad Autónoma de Madrid el habermeaceptado en su programa de doctorado.

Contents

1 Introduction 9

2 Theoretical preliminaries 132.1 Topological phases of matter . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 The SSH model . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Master equations . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Resolvent formalism . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Doublon dynamics 253.1 What are doublons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Doublon dynamics in 1D and 2D lattices . . . . . . . . . . . . . . . . . 29

3.2.1 Dynamics in the SSH chain . . . . . . . . . . . . . . . . . . . . 293.2.2 Dynamics in the 3 and Lieb lattices . . . . . . . . . . . . . . . 34

3.3 Doublon decay in dissipative systems . . . . . . . . . . . . . . . . . . . 403.3.1 Charge noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 Current noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.3 Experimental implications . . . . . . . . . . . . . . . . . . . . . 47

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.A E�ective Hamiltonian for doublons . . . . . . . . . . . . . . . . . . . . 513.B Bloch-Red�eld master equation . . . . . . . . . . . . . . . . . . . . . . 543.C Average over pure initial states . . . . . . . . . . . . . . . . . . . . . . 563.D Two-level system decay rates . . . . . . . . . . . . . . . . . . . . . . . 57

4 Topological quantum optics 594.1 Quantum emitter dynamics . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.1 Single emitter dynamics . . . . . . . . . . . . . . . . . . . . . . 614.1.2 Two emitter dynamics . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Single-photon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2.1 Scattering formalism . . . . . . . . . . . . . . . . . . . . . . . . 724.2.2 Scattering o� one and two emitters . . . . . . . . . . . . . . . . 74

4.3 Many emitters: e�ective spin models . . . . . . . . . . . . . . . . . . . 764.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8 CONTENTS

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.A Calculation of the self-energies . . . . . . . . . . . . . . . . . . . . . . 834.B Quantum optical master equation . . . . . . . . . . . . . . . . . . . . . 854.C Algebraic decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Conclusions and outlook 89

Bibliography 95List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

1Introduction

Topology is the �eld of mathematics that studies the properties of spaces that arepreserved under continuous transformations. It is not concerned about the particulardetails of those spaces, but on their most general aspects, like their number of connectedcomponents, the number of holes they have, etc. With such a broad point of view, itis not surprising that it has many applications in other sciences beyond mathematics.Its application to the �eld of condensed matter physics is relatively new. It beganaround the 1980s, when scientists such as Michael Kosterlitz, Duncan Haldane andDavid Thouless started using topological concepts to explain exotic features of newlydiscovered phases of matter, such as the quantized Hall conductance of certain 2Dmaterials at very low temperatures, the so-called integer Quantum Hall e�ect. In 2016they were awarded the Nobel Prize in physics for this and other works [1–4] whichopened the �eld of topological matter. Since then, the interest on this topic has grownexponentially, and so has the number of applications harnessing the exotic propertiesof topological phases.

Broadly speaking, topological matter is a new type of matter characterized by globaltopological properties. These properties stem, e.g., from the pattern of long-rangeentanglement in the ground state [5, 6] or, in the case of topological insulators andsuperconductors, from the electronic wavefunction in the whole Brillouin zone [7–9].Importantly, these new phases of matter display edge modes, which are conductingstates localized at the edges of the material. They are said to be topologically protected,that is, they are robust against perturbations which do not break certain symmetries ofthe system. For example, edge states in the integer quantum Hall e�ect are protectedagainst backscattering, unpaired Majorana fermions are protected against any pertur-bation that preserves fermion parity [10], etc. This makes topological phases of mattervery interesting for developing applications. Among them, perhaps the most exciting isthe realization of fault-tolerant quantum computers [11].

This new point of view in condensed matter physics puts forward many interestingchallenges. On a fundamental level, our understanding of topological phases of matteris not complete yet. How many di�erent phases are there? How to characterizethem? These questions have only been answered partially, mostly for systems of non-interacting particles [12]. On a practical level, we would like to predict which materials

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10 CHAPTER 1. INTRODUCTION

display topological properties and be able to probe them in experiment. To date, severaltopological materials have been demonstrated in experiments [13, 14], and systematicsearches have been carried out, unveiling that a large percentage of all known materialsare expected to have non-trivial topological properties [15–18].

While looking for real materials displaying topological properties is one possibility,an alternative is to simulate them in the lab. A quantum simulator is a device that canbe tuned in a way to mimic the behavior of another quantum system or theoreticalmodel that we want to investigate [19, 20]. Of course, a fully �edged quantum computerwould allow for the simulation any other quantum system [21]. But, we are yet farfrom building a proper, fault-tolerant, quantum computer. Thus, analogue quantumsimulation is a more reachable goal for the time being. One of the best platforms fordoing quantum simulation are ultracold atoms trapped in optical lattices [22, 23]. Inthese experiments an atomic gas of neutral atoms cooled to temperatures near absolutezero is loaded into a high vacuum chamber and trapped by dipolar forces at the maximaor minima of the electromagnetic �eld generated by interfering lasers. The opticalnature of the trapping potential allows for the generation of virtually any desired latticegeometry. Adding a periodic driving considerably enriches the physics of these systemsand provides a means for controlling and manipulating them. Such driving can producee�ects, such as coherent destruction of tunneling [24, 25], and can even be used todesign arti�cial gauge �elds [26–29]. Using these techniques some topological modelshave already been demonstrated in experiment [30–32].

Quantum dots (QD) are another candidate technology for doing quantum infor-mation processing [33–35], and quantum simulation [36, 37]. Laterally-de�ned QDsare made patterning electrodes on top of a sandwich of two semiconductors hosting atwo-dimensional electron gas (2DEG) at their interface. Applying particular voltagesto the electrodes, a speci�c potential landscape can be generated, which depletes the2DEG, trapping just a few electrons. In this manner, arti�cial atoms and molecules canbe created in solid-state devices [38], whose energy levels can be tuned by electrostaticmeans. Nowadays, increasingly larger arrays of QDs are being fabricated [39, 40], whichwould allow for the simulation of simple topological lattice models. Also hybrid systemswith improved capabilities are being created combining QDs with superconductingcavities [41].

Another research direction exports ideas from topological matter to other �elds ofphysics. For example, topological insulators can be used as a guide for the design ofnovel metamaterials with interesting mechanical, acoustic and optical properties [42,43]. In particular, the application of topological ideas to photonics has given rise to anew �eld known as topological quantum optics [44]. This is a rather young �eld wheremany open questions are waiting to be answered.

The outline of this thesis is as follows: In chapter 2 we review di�erent mathematicaltechniques that we use in subsequent chapters; minor details are left as appendices

11

at the end of each chapter. In chapter 3 we investigate the dynamics of interactingfermions in driven topological lattices. In chapter 4 we explore the dynamics of quantumemitters coupled to a 1D topological photonic lattice, namely a photonic analogue ofthe SSH model. Last, we summarize our �ndings and give an overview of the possibleexperimental implementations in chapter 5.

2Theoretical preliminaries

In an attempt to keep the text as much self-contained as possible, here we brie�yintroduce the most important theoretical tools used to derive the results presented inthis thesis. First, we comment on topological matter, placing special emphasis on theSSH model, a key player in subsequent chapters. Second, we give a primer on Floquettheory, a basic tool for studying driven systems with some periodic time dependence;we will use it in section 3.2. Last, we discuss two di�erent approaches to open quantumsystems: master equations and resolvent operator techniques. They are relevant forsection 3.3 and the whole of chapter 4.

2.1. Topological phases of matter

Towards the middle of the last century, Landau proposed a theory for matter phasesand phase transitions where di�erent phases are characterized by symmetry breakingand local order parameters. Then, the discovery of the integer [45] and fractional [46]Quantum Hall e�ects in the 1980s revolutionized the �eld of condensed matter physics.These new phases could not be understood in terms of local order parameters, andposed a problem for the established theory. After that, many other phases that fallbeyond Landau’s paradigm have been discovered and a new picture has started toemerge, one where topology plays a major role [5].

Topological phases can be divided in two large families: topologically ordered phasesand symmetry-protected topological phases (SPT). Both refer to gapped phases, i.e., witha nonzero energy gap above the ground state in the thermodynamic limit at zero tem-perature. The fundamental classifying principle that operates in all this discussion is asfollows: Two systems belong to the same phase if one can be deformed adiabatically intothe other without closing the gap, and they belong to di�erent phases otherwise. Phaseswith true topological order are long-range entangled with a degenerate ground statewhose degeneracy depends on the topology of the underlying space, and quasiparticleswith fractional statistics called anyons. Examples include the fractional Quantum Halle�ect [47], chiral spin liquids [48], or the toric code [49], to name a few. SPT phases,on the other hand, are short-range entangled and they require the presence of certain

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14 CHAPTER 2. THEORETICAL PRELIMINARIES

symmetries to be well-de�ned. A characteristic of all topological phases is the presenceof edge modes or excitations that are said to be “topologically protected”, meaning thatthey are robust against certain types of disorder.

SPT phases can be further subdivided into interacting and non-interacting phases,depending on whether particles interact with each other or not. Examples of the �rsttype are Haldane’s spin-1 chain and the AKLT model [50]. Examples of the second typeinclude Kitaev’s chain [10] and Chern insulators. Although a complete classi�cationof all SPT phases is not known yet, some classi�cation schemes have been elucidatedfor particular cases. For example, 1D interacting phases can be characterized by theprojective representations of their symmetry groups [51–53], and a complete classi�-cation of non-interacting fermionic phases, also known as topological insulators andsuperconductors, has been obtained in terms of a topological band theory, which assignstopological invariants to the energy bands of their spectrum [7–9]. For their classi�-cation the relevant symmetries are [12]: Time-reversal symmetry, which correspondsto an antiunitary operator that commutes with the Hamiltonian, charge-conjugation(also known as particle-hole), which corresponds to an antiunitary operator that an-ticommutes with the Hamiltonian, and chiral (also known as sublattice) symmetry,which corresponds to a unitary operator that anticommutes with the Hamiltonian. Thepresence of these symmetries imposes restrictions on the single-particle Hamiltonianmatrix, H , as shown in the table below

Symmetry De�nition

time-reversal U H ∗U † = H

charge-conjugation UH ∗U † = −H

sublattice UHU † = −H

where U� , � ∈ { ,,}, are unitary matrices and “∗” denotes complex conjugation.Note that having two of them, automatically implies that all three symmetries arepresent. As it turns out, there are ten di�erent classes depending on whether theaforementioned symmetries are present or absent and whether they square to ±1 ifpresent. For each class, the classi�cation scheme determines how many distinct phasesare possible depending on the dimensionality of the system. There are essentially threepossibilities: there may be just the trivial phase; two phases, one trivial and anothertopological, distinguished by a ℤ2 topological invariant; or there may be in�nitely manydistinct phases distinguished by a ℤ topological invariant.

A word of caution is in order. Throughout the literature, the term “topological” isused somewhat vaguely. In many cases it is just a label to refer to any phase other thanthe trivial. But, what is a trivial phase? Well, it is just a phase that can be connectedadiabatically with that of the vacuum, or the phase of a disconnected lattice.

2.1. TOPOLOGICAL PHASES OF MATTER 15

2.1.1. The SSH modelOne of the simplest models featuring a non-interacting SPT phase is the SSH model,named after Su, Schrie�er and Heeger, who �rst studied it in the 1970s [54, 55]. Itdescribes non-interacting particles hopping on a 1D lattice with staggered nearest-neighbour hopping amplitudes J1 and J2. The lattice consists of N unit cells, each onehosting two sites that we label A and B, see Fig. 2.1(a). Its Hamiltonian can be writtenas

HSSH = −∑j(J1c†jAcjB + J2c†j+1AcjB + H.c.) , (2.1)

where cj� annihilates a particle (boson or fermion) in the � ∈ {A, B} sublattice atthe jth unit cell. The hopping amplitudes are usually parametrized as J1 = J (1 + �)and J2 = J (1 − �), where � ∈ [−1, 1] is the so-called dimerization constant. Assumingperiodic boundary conditions, the Hamiltonian can be written in momentum space asHSSH = ∑k V †

k HkVk , with Vk = (ckA, ckB)T , and

Hk = (0 f (k)

f ∗(k) 0 ) , f (k) = −J (1 + �) − J (1 − �)e−ik . (2.2)

Here, f (k) denotes the coupling between the modes ck� = ∑j e−ikjcj� /√N . This Hamilto-

nian can be easily diagonalized as

HSSH =∑k!k (u†kuk − l†k lk) , (2.3)

with

!k = |f (k)| = J√2(1 + �2) + 2(1 − �2) cos(k) , (2.4)

uk =1√2 (e−i�kckA + ckB) , lk =

1√2 (e−i�kckA − ckB) , (2.5)

and �k = arg(f (k)). Its spectrum consists of two bands with dispersion relation −!k(lower band), and !k (upper band), spanning the ranges [−2J , −2|� |J ] and [2|� |J , 2J ]respectively, see Fig. 2.1(b).

The SSH model has all three symmetries mentioned in previous paragraphs: time-reversal, charge-conjugation and chiral symmetry. Thus, according to the classi�cation,it belongs to the BDI class, which in 1D supports distinct topological phases char-acterized by a ℤ topological invariant. To �nd this invariant, let us point out thatchiral symmetry is represented in momentum space by the z-Pauli matrix �z , that is,�zHk�z = −Hk . Expressing Hk in the basis of Pauli matrices, Hk = ℎ0(k)I + h(k) ⋅ �� , thissymmetry constraint forces ℎ0(k) = 0 and ℎz(k) = 0, so the vector h(k) lies on a plane.Also, the existence of a gap requires h(k) ≠ 0. Therefore, h(k) is a map from S1 (the �rst


Figure 2.1: (a) Schematic drawing of the SSH model: a 1D lattice with two sites per unitcell (in gray), labelled A and B, with alternating hopping amplitudes J1 = J (1 + �) andJ2 = J (1 − �). (b) Upper and lower energy bands of the SSH model.

Brillouin zone) to ℝ2 ⧵ {0}. Such maps can be characterized by a topological invariantthat only takes integer values: the winding number, , of the curve h(k) around theorigin. Furthermore, as long as symmetries are preserved it is impossible to change thewinding number of the curve without making it pass through the origin, in accordancewith the fact that distinct topological phases are separated by phase transitions in whichthe gap closes.

The SSH model can be in two distinct phases: a topological phase with = 1, forJ1 < J2 (� < 0), or a trivial phase with = 0, for J1 > J2 (� > 0). We remark that higherwinding numbers can be achieved if longer-range hoppings are included [1*, 2*]. Bythe bulk-boundary correspondence, the value of | | corresponds to the number of pairsof edge states supported by the system [56]. These states are exponentially localizedat the edges of the chain and its energy is pinned on the middle of the band gap. InFig. 2.2(a) we show the energy spectrum of a �nite chain consisting of N = 10 dimers.There, it can be seen how for � < 0 the energies of two states detach from the bulkenergy bands (shaded areas) and quickly converge to zero. By the energy spectrum, inFig. 2.2(b), we show the amplitudes of the two midgap states for a particular value of� < 0, proving that they are exponentially localized to the edges of the chain. To betterunderstand the features of the energy spectrum and the edge states let us turn again to

2.1. TOPOLOGICAL PHASES OF MATTER 17

chiral symmetry. For systems de�ned on a lattice, we say that the system is bipartiteif the lattice can be divided in two sublattices (A and B) such that hopping processesonly connect sites belonging to di�erent sublattices. As it turns out, any bipartitelattice has chiral symmetry embodied in the transformation cjA → cjA, cjB → −cjB.Its action over a single-particle wavefunction is to reverse the sign of the amplitudeson one sublattice—hence, the name sublattice symmetry—which changes the sign ofthe Hamiltonian. This implies that the eigenvalues either come in pairs with oppositeenergies or have energy equal to zero and are also eigenvalues of the chiral symmetryoperator. Note that the eigenstates in a pair have the same wavefunction, except for achange of the sign of the amplitudes in one sublattice. On the other hand, eigenstateswith zero energy have support on a single sublattice. Edge states in the thermodynamiclimit have zero energy, but in a �nite system they hybridize forming symmetric andantisymmetric combinations which constitute a chiral symmetric pair with an energysplitting Δ� that decreases exponentially with increasing chain size, Δ� ∝ e−N /�.

Figure 2.2: (a) Single-particle spectrum as a function of the dimerization constant for a�nite chain with N = 10 unit cells. The shaded areas correspond to the ranges of theenergy bands in the thermodynamic limit (b) Edge states for the same chain at � = −0.2;blue (red) bars correspond to the amplitudes in sublattice A (B)

For chains with an odd number of sites there must be an odd number of single-particle states precisely at zero energy due to chiral symmetry. Indeed, they supportonly one state at zero energy, exponentially localized on a single edge, with weight onjust one sublattice. Note that in this case, changing the sign of � amounts to invertingthe chain spatially. Thus, depending on the sign of � this edge state is localized on theleft or right edge.


2.2. Floqet theoryFloquet theory is concerned with the solution of periodic linear di�erential equations.In quantum mechanics, it is used to solve systems modelled by Hamiltonians with someexplicit periodic time dependence, that is, solutions of the Schrödinger equation

i)t | (t)⟩ = H (t)| (t)⟩ (2.6)

with H (t + T ) = H (t) (here and throughout the text we work in units such that ℏ = 1).For simplicity, we will just consider systems with a Hilbert space of �nite dimension.Floquet’s theorem states that there is a fundamental set of solutions {| � (t)⟩} to thisequation of the form | � (t)⟩ = e−i�� t |�� (t)⟩, with �� real and |�� (t)⟩ periodic with thesame periodicity as the Hamiltonian, |�� (t + T )⟩ = |�� (t)⟩ [57]. Therefore, any solutionof Eq. (2.6) can be expanded as

| (t)⟩ =∑�c�e−i�� t |�� (t)⟩ , (2.7)

with time independent coe�cients c� . In analogy with Bloch’s Theorem, �� is calledquasienergy and |�� (t)⟩ is called Floquet mode. Introducing | � (t)⟩ in (2.6), we can seethat Floquet modes satisfy the eigenvalue equation

|��⟩ ≡ (H − i)t) |��⟩ = �� |��⟩ , (2.8)

where is the so-called Floquet operator, and it is Hermitian.Equivalently, Floquet’s theorem states that for time-periodic systems the unitary

time evolution operator can be factorized as

U (t2, t1) =∑�e−i�� (t2−t1)|�� (t2)⟩⟨�� (t1)| = e−iHeff (t2−t1)P (t2, t1) , (2.9)

with Heff = ∑� �� |�� (t2)⟩⟨�� (t2)| being an e�ective time-independent Hamiltonian, andP (t2, t1) = ∑� |�� (t2)⟩⟨�� (t1)| a unitary operator, T -periodic in both of its arguments.The long-term dynamics is governed by Heff , wile the dynamics within a period, alsoknown as the micromotion, is given by P . Note that there is a gauge freedom and wecould also have written

U (t2, t1) = P†(t0, t2)e−iHeff (t2−t1)P (t0, t1) , (2.10)

setting Heff = ∑� �� |�� (t0)⟩⟨�� (t0)|. In some texts the micromotion operator is writtenas P (t0, t) = eiK (t), with K (t) Hermitian and T -periodic, depending implicitly on thechoice of t0.

Floquet modes and quasienergies play a central role in the study of periodic sys-tems. There are several ways to compute them. One can integrate numerically the

2.2. FLOQUET THEORY 19

dynamics of some states forming an orthonormal basis at t1 = 0, obtaining U (t, 0) fort ∈ [0, T ]. The eigenvalues of U (T , 0) are the complex phases e−i��T , with associatedeigenvectors |�� (T )⟩ = |�� (0)⟩. Their full time dependence can be reconstructed as|�� (t)⟩ = ei�� tU (t, 0)|�� (0)⟩. Note that, since the quasienergies appear in a complexphase, they are de�ned modulo 2�/T = !. Given a quasienergy �� , we can add toit a multiple of the frequency ��n = �� + n! and the corresponding Floquet mode|��n(t)⟩ = ein!t |�� (t)⟩ will also satisfy Eq. (2.8) with eigenvalue ��n. In fact any of theseFloquet modes, |��n(t)⟩, n ∈ ℤ, yields the same solution to the Schrödinger equation.So, the quasienergy spectrum is periodic, much like quasimomentum in crystallinesolids, and it su�ces to consider the quasienergies within a range of width !.

Another procedure exploits the fact that that Floquet modes are periodic in time, sothey can be expanded in Fourier series,

|��⟩ =∑ne−in!t |c�n⟩ , |c�n⟩ =

1T ∫

T

0dt ein!t |�� (t)⟩ . (2.11)

The time-independent vectors |c�n⟩ can themselves be expanded into some basis, {|�⟩},of the system’s Hilbert space. Now, we can identify e−in!t |�⟩ ≡ |n�(t)⟩ as basis states ofan enlarged Hilbert space built as the product of the system’s Hilbert space and thespace of T -periodic functions [58]. We will denote the elements of this space as |�⟩⟩, sothat in “time representation” they are ⟨t |�⟩⟩ = |�(t)⟩. An inner product can be de�nedin this enlarged space as the composition of the usual inner products in each of theconstituent spaces

⟨⟨�|�′⟩⟩ = 1T ∫

T

0dt ⟨�(t)|�′(t)⟩ . (2.12)

With respect to this basis, the matrix elements of the Floquet operator are

⟨⟨n′�′||n�⟩⟩ = 1T ∫

T

0dt ein′!t⟨� |H (t) − i)t |�⟩e−in!t

= ⟨�′|Hn′−n|�⟩ − n!�n′n��′� , (2.13)

where Hn′−n is the (n′ − n)th Fourier component of the Hamiltonian. Thereby, wehave transformed a time-dependent problem into a time-independent one with anextra (in�nite-dimensional) degree of freedom. This extra degree of freedom is directlyrelated to the apparent redundancy of the Floquet modes mentioned earlier. Frequently,a time dependence in the Hamiltonian stems from the coupling to the modes of, e.g.,a laser �eld in the semiclassical limit [59]. In this limit the basis states |�n⟩⟩ can beinterpreted as having a de�nite number of photons, and Floquet modes can be viewedas dressed states. Based on this analogy, the indices (n′, n) in Eq. (2.13) are referredto as the photon indices. Furthermore, the matrix elements of Hm are said to describem-photon processes.


The bene�t of this technique is that it allows the direct application of methods usedfor the diagonalization of time-independent Hamiltonians to compute quasienergies andFloquet modes. In the high frequency regime, blocks with di�erent photon number arefar apart in energy, and it makes sense to use perturbation theory to block diagonalizethe Floquet operator [60–62]. This provides a series expansion of Heff in powers of !−1,known as the high-frequency expansion (HFE). In some situations, computing a fewterms in this expansion is easier than computing the Floquet modes and quasienergies,and provides more physical insight. In section 3.2, we use this technique to investigatethe dynamics of a pair of strongly-interacting fermions under the action of an ac �eld.

2.3. Open qantum systems

Quantum mechanics has allowed us to explain many interesting phenomena at thecost of a more complicated description of fundamental particles and interactions. Forexample, the number of variables needed to describe the state of an ensemble of particlesgrows exponentially with the number of particles in the ensemble. This makes it veryhard to analyze large systems involving many particles, or systems that interact withexternal, uncontrolled degrees of freedom. However, �nding ways to tackle theseproblems is a necessity, since more often than not this is the situation we face in realexperiments.

As opposed to closed quantum systems, open quantum systems are systems thatinteract with an environment, also called bath or reservoir, which is a collection ofin�nitely many degrees of freedom. Through this interaction the system exchangesinformation, energy and particles with the environment. As a result, dissipation anddecoherence are introduced into the system [63]. The former is the phenomenonby which the system exchanges energy with the environment, eventually reachingthermal equilibrium with it, while the latter is the phenomenon by which coherentsuperpositions of states are lost over time. Below we summarize two distinct approachesto the study of open quantum systems.

2.3.1. Master eqationsA proper description of the system taking into account the e�ects of decoherence anddissipation is given by a density matrix, which besides pure states also includes statisticalmixtures of them. The usual way to study this type of problems involves tracing outthe bath degrees of freedom, obtaining a �rst order linear di�erential equation for thesystem’s reduced density matrix, the so-called master equation. There are di�erentways to do this depending on the regime and approximations that apply to the systemunder consideration [64].

2.3. OPEN QUANTUM SYSTEMS 21

We now proceed to show how to obtain a master equation, valid in the regime ofweak system-bath coupling. The combination of system and bath is described as awhole by the Hamiltonian H = HS +HB +HI , where HS and HB are the free Hamiltoniansof system and bath respectively, and HI is the Hamiltonian that describes the interactionbetween them. We restrict the interaction to the case where HI is linear in both systemand bath operators, {Xj} and {Bj} respectively, HI = ∑j Xj ⊗ Bj . The entire systemevolves according to the von-Neumann equation:

�(t) = −i[HI (t), �(t)] , (2.14)

Here, � is the full density matrix of the system plus the bath; the tilde over an operatordenotes the interaction picture,

O(t) ≡ ei(HS+HB)tOe−i(HS+HB)t , (2.15)

where O is the corresponding operator in the Schrödinger picture. Notice that at timet = 0, both the Schrödinger and the interaction picture representations coincide. Theintegral form of Eq. (2.14) is

�(t) = �(0) − i ∫t

0ds [HI (s), �(s)] . (2.16)

Inserting this expression for �(r) back in the right hand side of Eq. (2.14), tracing overthe bath degrees of freedom, we get

�S(t) = −i trB[HI (t), �(0)] − ∫t

0ds trB[HI (t), [HI (s), �(s)]] . (2.17)

To obtain a closed equation for the reduced density matrix of the system, we need tomake some approximations:

• Born approximation: We assume that at all times, the total density matrix canbe factorized as �(t) ≈ �S(t) ⊗ �B. This amounts to neglect any entanglementbetween the system and the bath, which is justi�ed if the coupling between themis small enough. Furthermore we will always consider a thermal state for thebath �B ∝ e−�HB .

• Markov approximation: We replace �S(s) by �S(t) in the integrand, such that thetime evolution of the system only depends on its current state but not on itsprevious states. Furthermore, we let the lower integration bound go to −∞ andmake a change of variable s = t −� . This approximation is justi�ed if the timescalein which the reservoir correlations decay is small compared to the timescale inwhich the system varies noticeably.


Last, if ⟨Bj(t)⟩ = 0, where we denote ⟨x⟩ ≡ trB(x�B), as is the case in all the modelspresented in this thesis, we can neglect the �rst term, obtaining

�S(t) = − ∫∞

0d� trB[HI (t), [HI (t − � ), �S(t) ⊗ �B]] . (2.18)

Transforming it back to the Schrödinger picture, we get

�S = −i[HS , �S] − ∫∞

0d� trB[HI , [HI (−� ), �S ⊗ �B]]

≡ −i[HS , �S] + [�S] , (2.19)

where the �rst term accounts for the coherent unitary dynamics, whereas the secondincludes dissipation and decoherence.

This equation, known in the literature as the Bloch-Red�eld master equation [65],is the one we will use for studying the decay of doublons in noisy environments, seesection 3.3 and appendix 3.B. Note that this equation does not necessarily preserve thepositivity of the density matrix. One has to perform a further rotating-wave approxima-tion, which involves averaging over the rapidly oscillating terms in Eq. (2.18) to obtainan equation that does preserve it, i.e., one that can be put in Lindblad form [64]. Thisother equation, known as the quantum optical master equation in some contexts, is theone we will use for studying the dynamics of quantum emitters coupled to a topologicalbath , see chapter 4 and appendix 4.B.

2.3.2. Resolvent formalismQuite often in quantum physics we want to know what happens to a particular sys-tem of interest (e.g. a qubit or atom) when coupling it to an external driving, or abath. The coupling may induce transitions between the bare eigenstates of the system,whose probability can be computed exactly in some cases with resolvent operatortechniques [66, 67].

Let us consider a system with free Hamiltonian H0, perturbed such that the actualHamiltonian of the system is H = H0 + V . The time evolution operator satis�es

i)tU (t, t0) = HU (t, t0) . (2.20)

We can split the time evolution operator into two operators G±(t, t0) that evolve thestate of the system at time t = t0 forwards and backwards in time respectively,

G±(t, t0) = ±U (t, t0)Θ (±(t − t0)) . (2.21)

Here, Θ(t) denotes Heaviside’s step function. Di�erentiating Eq. (2.21) we can see thatthey satisfy the same equation,

(i)t − H )G±(t, t0) = i�(t − t0) . (2.22)

2.3. OPEN QUANTUM SYSTEMS 23

They are usually referred to as the retarded (“+”) and advanced (“−”) Green’s functionsor propagators. For a time-independent system G±(t, t0) depends only on � = t − t0. Letus now introduce their Fourier transform

G±(� , 0) = − 12�i ∫

∞

−∞dE e−iE�G±(E) , (2.23)

G±(E) = 1i ∫∞

−∞d� eiE�G±(� , 0) , (2.24)

Substituting G+(� , 0) = e−iH�Θ(� ) in Eq. (2.24) we �nd

G+(E) = 1i ∫∞

0d� ei(E−H )� = lim

�→0+1i ∫

∞

0d� ei(E−H+i�)�

= lim�→0+

1E − H + i� , (2.25)

and similarly,

G−(E) = lim�→0+

1E − H − i� . (2.26)

These expressions suggest the de�nition of the operator

G(z) = 1z − H , (2.27)

which is a function of a complex variable z, such that G±(E) = G(E ± i0+). G(z) is calledthe resolvent of the Hamiltonian H .

From the de�nition (2.21), it is clear that the transition amplitudes from an initialstate |�⟩ to a �nal state |�⟩ after a period � can be computed as

⟨� |U (� )|�⟩ = − 12�i ∫

∞

−∞dE e−iE�⟨� |G(E + i0+)|�⟩ . (2.28)

Thus, the analytical properties of the matrix elements of the resolvent play a crucial rolein determining the dynamics of the system. It can be shown that the matrix elementsof G(z) are analytic in the whole complex plane except for the real axis, where theyhave poles and branch cuts at the discrete and continuous spectrum of H respectively.Furthermore, it is possible to continue analytically G(z) bridging the cuts in the realaxis, exploring other Riemann sheets of the function, where it may no longer be analyticand may contain poles with a nonzero imaginary part, the so-called unstable poles.

We will now see how to obtain explicit formulas for the relevant matrix elementsof the resolvent. Suppose we want to know what happens to the states of a particularsubspace spanned by {|�⟩}, which are eigenstates ofH0. Let us denote the projector onto


that subspace as P = ∑� |�⟩⟨� |, and the projector on the complementary subspace asQ = 1 − P . Then, from the de�ning equation of the resolvent (z − H )G = 1, multiplyingit from the right by P , and from the left by P and Q, we get the equations

P (z − H )P [PGP] − PVQ [QGP] = P , (2.29)−QVP [PGP] + Q(z − H )Q [QGP] = 0 . (2.30)

Solving for QGP in Eq. (2.30)

QGP = Qz − QH0Q − QVQVPGP , (2.31)

and substituting back in Eq. (2.29), we obtain

P [z − H0 − V − V Qz − QH0Q − QVQV ] PGP = P . (2.32)

Introducing the operator

R(z) = V + V Qz − QH0Q − QVQV , (2.33)

which is known as the level-shift operator, we can express

PG(z)P = Pz − PH0P − PR(z)P

. (2.34)

From Eq. (2.31) we can now obtain

QG(z)P = Qz − QH0Q − QVQV

Pz − PH0P − PR(z)P

. (2.35)

In section 4.1 and appendix 4.A we use these formulas for computing the survivalprobability amplitude of the excited state of one and two quantum emitters.

3Doublon dynamics

Recent experimental advances have provided reliable and tunable setups to test andexplore the quantum mechanical world. Paradigmatic examples are ultracold atomstrapped in optical lattices [22, 23], quantum dots [34, 35, 68, 69], and photonic crys-tals [70–73]. In these setups, quantum coherence is responsible for many exotic phe-nomena, in particular, the transfer of quantum information between di�erent locations,a process known as quantum state transfer. Even particles themselves, which may carryquantum information encoded in their internal degrees of freedom, can be transferredin a controlled manner in these setups. Given its importance in quantum informationprocessing applications, many theoretical and experimental works have studied theseprocesses in recent years [74–78].

Floquet engineering, that is, the use of periodic drivings in order to modify theproperties of a system, has become an essential technique in the cold-atom toolbox,which has enabled the simulation of some topological models [30–32]. However, mostof the studies carried out so far utilize this technique aimed at the single-particle level.In this chapter we investigate the dynamics of two strongly interacting fermions boundtogether, forming what is termed a “doublon”, under the action of periodic drivings.We show how to harness the topological properties of di�erent lattices to transferdoublons [3*], and demonstrate a phenomenon by which the driving con�nes doublondynamics to a particular sublattice [4*]. Afterwards, we address the question whetherdoublons can be observed in noisy systems such as quantum dot arrays [5*].

3.1. What are doublons?

An ubiquitous model in the �eld of Condensed Matter is the Hubbard model. Despite itsseeming simplicity, it captures a great variety of phenomena ranging from metallic be-havior to insulators, magnetism and superconductivity. The Hamiltonian of this modelconsists of two contributions: a hopping term HJ that corresponds to the kinetic energyof particles moving in a lattice, and an on-site interaction term HU that corresponds tothe interaction between particles occupying the same lattice site. For spin-1/2 fermions,

25

26 CHAPTER 3. DOUBLON DYNAMICS

this Hamiltonian can be written as

H = − ∑i, j, �

Jijc†i�cj� + U ∑ini↑ni↓ ≡ HJ + HU . (3.1)

Here, ci� annihilates a fermion with spin � ∈ {↑, ↓} at site i, and ni� = c†i�ci� is the usualnumber operator. Jij is the, possibly complex, hopping amplitude between sites i andj (Hermiticity of H requires Jij = J ∗ji). The interaction strength U corresponds to theenergy cost of the double occupancy.

As we shall see, in the strongly interacting limit of the Hubbard model (U ≫|Jij |) particles occupying the same lattice site can bind together, even for repulsiveinteractions. This happens due to energy conservation, and the fact that, in a lattice,the maximum kinetic energy a particle can have is limited to the width of the energybands. Therefore, an initial state where the particles occupy the same site cannot decayto a state where the particles are separated, since they would not have enough kineticenergy on their own to compensate for the large interaction energy. In principle, bothbosons [79–81] and fermions [82, 83] can form such N -particle bound states. Whilethe former allow for any occupation number, for fermions with spin s the occupationof one site is restricted to at most 2s + 1 particles. In particular, two spin-1/2 fermionsmay be in a singlet spin con�guration on the same lattice site and form a doublon. Overthe last years they have been investigated experimentally, mostly with cold atoms inoptical lattices [84–88].

The Hilbert space of two particles in a singlet con�guration is spanned by two typesof states: single-occupancy states

1√2 (c†i↑c†j↓ − c†i↓c†j↑) |0⟩ , 1 ≤ i < j ≤ N , (3.2)

and double-occupancy states, also known as doublons,

c†j↑c†j↓|0⟩ , j = 1,… , N . (3.3)

Both are eigenstates of HU with eigenvalues 0 and U respectively. The hopping termcouples both types of states, so that they no longer are eigenstates of the full Hamiltonian.However, for su�ciently large values of U the eigenstates also discern in two groups,namely, N (N − 1)/2 states with energies |�n| ≲ 4J , which have a large overlap withthe single-occupancy states, and N states with energies |�n| ≃ U , which have a largeoverlap with the doublon states. We will refer to the span of the former as the low-energy subspace (they are also referred to as scattering eigenstates), and the span ofthe latter as the high-energy subspace (also known as two-particle bound states). Thisdistinction can be clearly appreciated in Fig. 3.1. In this regime, a state initially havinga high double occupancy will remain like that as it evolves in time. In this sense, we

3.1. WHAT ARE DOUBLONS? 27

can say that the total double occupancy is an approximate conserved quantity in thestrongly-interacting regime.

Treating the tunneling as a perturbation it is possible to obtain an e�ective Hamil-tonian for the high-energy subspace [89]. The method, which goes by the name ofSchrie�er-Wol� transformation [90], provides a unitary transformation that block-diagonalizes the Hamiltonian perturbatively in blocks of states with di�erent numberof doubly occupied sites. To achieve this, we �rst split the kinetic term into hoppingsthat increase, decrease, and leave unaltered the double occupancy, HJ = H +

J + H −J + H 0

J ,where

H +J = − ∑

i, j, �Jijni�c†i�cj�ℎj� , (3.4)

H −J = − ∑

i, j, �Jijℎi�c†i�cj�nj� , (3.5)

H 0J = − ∑

i, j, �Jij (ni�c†i�cj�nj� + ℎi�c†i�cj�ℎj�) . (3.6)

In these equations � denotes the opposite value of � and ℎi� ≡ 1−ni� . The transformationof H by any unitary eiS (S† = S) can be computed as

H = eiSHe−iS = H + [iS, H ]1! + [iS, [iS, H ]]2! +⋯ . (3.7)

Noting that [HU , H �J ] = �UH �

J , � ∈ {±, 0}, one can readily see that in order to removethe terms that couple the two sectors at zeroth order, H ±

J , one has to choose iS =(H +

J − H −J ) /U . Then,

H = H 0J + HU + [H +

J , H −J ] + [H 0

J , H −J ] + [H +

J , H −J ]

U + O (U −2) . (3.8)

Keeping terms up to order U −1 that act non-trivially on doublon states, we obtain thefollowing e�ective Hamiltonian for doublons

Heff = HU +1U H +

J H −J . (3.9)

Since we are concerned with states with no singly occupied sites, in H +J H −

J we haveto consider just those hopping processes where a doublon splits and then recombinesagain to the same site or to an adjacent site,

H +J H −

J∗= ∑i, j, �

|Jij |2ni�c†i�cj�ℎj�ℎj�c†j�ci�ni� + ∑i, j, �

J 2ijni�c†i�cj�ℎj�ℎi�c†i�cj�nj� . (3.10)

The asterisk on top of the equal sign denotes equality when restricted to the doublonsubspace. The terms in the �rst sum can be rewritten as

ni�c†i�cj�ℎj�ℎj�c†j�ci�ni� = ni�ni� − ni�ni�nj�nj� , i ≠ j . (3.11)


High-energy subspace

Low-energy subspace

Figure 3.1: Energy spectrum of H and Heff for two spin-1/2 fermions forming a singletin the SSH-Hubbard model (chain with N = 3 dimers and � = −0.2, see next section) asa function of the interaction strength. For large interactions, the e�ective Hamiltoniancorrectly reproduces the eigenenergies of the high-energy subspace.

Here, we have used the fact that for a doublon state, if a site is occupied it has to bedouble occupied, i.e., ni� ∗= ni�ni� ∗= ni� . As for the terms in the second sum,

ni�c†i�cj�ℎj�ℎi�c†i�cj�nj� = c†i�c†i�cj�cj� , i ≠ j . (3.12)

Finally, using doublon creation and annihilation operators, d†i = c†i↑c†i↓ and di = ci↓ci↑, thee�ective Hamiltonian for doublons can be written as

Heff =∑i, j

2J 2ijU d†i dj +∑

i�id†i di −∑

i, j

2|Jij |2U d†i did†j dj , (3.13)

with �i = U + ∑j 2|Jij |2/U . The �rst term corresponds to an e�ective hopping fordoublons, the second to an e�ective on-site chemical potential, and the third to anattractive interaction between doublons.

We remark that for a 1D lattice with homogeneous nearest-neighbor hoppings, thetwo-body spectrum can be computed exactly with Bethe Ansatz techniques [91].

In appendix 3.A we derive an e�ective Hamiltonian for doublons including also thee�ect of an external periodic driving.

3.2. DOUBLON DYNAMICS IN 1D AND 2D LATTICES 29

3.2. Doublon dynamics in 1D and 2D latticesAfter understanding what doublons are, and what makes them stable quasiparticles,we are now in a good position to discuss their dynamics. In this section we show howthe interplay between topology, interactions, and driving, leads to surprising e�ectsthat constrain the motion of doublons.

3.2.1. Dynamics in the SSH chainInterestingly, the topological properties of the SSH model can be harnessed to producethe transfer of non-interacting particles between the ends of a �nite chain, withoutthem occupying the intervening sites. Key to this process is the presence of edge states.Let us recall their properties. A �nite dimer chain supports two edge states |ES±⟩, withenergies ±�/2, when it is in the topological phase (� < 0). They are well separatedenergetically from the rest of (bulk) states, and there is a small energy splitting betweenthem that decreases exponentially with increasing chain size, � ∝ e−N /�. Each of themis exponentially localized on both edges of the chain, and they are even and odd underspace inversion; thus, they can be regarded as a non-local two level system. As aconsequence, a particle in a superposition of both edge states will oscillate between theends of the chain as it evolves in time. For example, if the particle is initially on the�rst site of the chain | (0)⟩ = |1⟩ ≃ (|ES+⟩ + |ES−⟩) /

√2, the probability to �nd it on the

last site of the chain |2N ⟩ ≃ (|ES+⟩ − |ES−⟩) /√2 is

|⟨2N | (t)⟩|2 = 14||e−i�t/2 − ei�t/2||

2 = 1 − cos(�t)2 , (3.14)

thus, in a time period T0 = �/� the particle will be transferred with certainty to theother end of the chain.

We want to know how interaction between particles a�ects this process, and seewhether the controlled transfer of doublons is possible. The Hamiltonian of the systemcorresponds to Eq. (3.1) with

Jij ={J [1 + �(−1)max(i,j)] , |i − j | = 10 , otherwise

. (3.15)

This is just a di�erent way to express HSSH [Eq. (2.1) in section 2.1] including the spindegree of freedom and the on-site interaction between particles. We will refer to thismodel as the SSH-Hubbard model.

In Fig. 3.2 (top row) we plot the dynamics of a doublon starting on the �rst site of asmall chain. As can be observed, the edge-to-edge oscillations are lost in the strongly-interacting regime. We can understand why looking at the e�ective Hamiltonian for


doublons

Heff =∑j(J eff1 d†2jd2j−1 + J eff2 d†2j+1d2j + H.c.) +∑

j�jd†j dj , (3.16)

which can be obtained directly from Eq. (3.13), neglecting the interaction betweendoublons, since we are considering just one of them. The e�ective hopping amplitudesare J eff1 = 2J 2(1 + �)2/U and J eff2 = 2J 2(1 − �)2/U . Importantly, the e�ective local chemicalpotential �j is di�erent for the ending sites, as they have one fewer neighbor,

�j ={�bulk = J eff1 + J eff2 + U , 2 ≤ j ≤ 2N − 1�edge = J eff1 + U , j = 1, 2N . (3.17)

This chemical potential di�erence at the edges spoils the chiral symmetry of the model,and is able to shift the energy of the edge states that appear in the topological phaseof the unperturbed SSH model. In fact, for the particular value Δ� ≡ �bulk − �edge = J eff2 ,they completely merge into the bulk bands, see Fig. 3.2 (top row). This explains whythe mechanism that produces the edge-to-edge oscillations in the single particle casedoes not apply directly to the doublon case.

We can now think of di�erent ways to restore the edge states in order to producethe transfer of doublons. One possibility is is to add a local potential at the edges of thechain so as to compensate for the di�erence in chemical potential,

H = HJ + HU + Vgate∑�(n1� + n2N� ) , (3.18)

with Vgate = J 2(1 − �)2/U . The resulting e�ective Hamiltonian has a homogeneouschemical potential �j = �gate for all j, and is formally identical to that of the SSH model.Indeed, this produces the desired dynamics, see Fig. 3.2 (middle row). Another possibilityis to take advantage of this chemical potential di�erence, which can localize states onthe edges of the chain of the Shockley type. This requires the hopping amplitudes tobe smaller than Δ�. Usually this is not the case, however there is an e�cient way toinduce such states by driving the system with a high-frequency ac �eld. The ac �eldrenormalizes the hoppings, which become smaller than in the undriven case [24, 25,92]. This cannot be achieved, for example, by simply reducing the hoppings J1 and J2by hand, since this will also a�ect the e�ective chemical potential which still will be ofthe same order of J eff1 and J eff2 . To model the ac �eld we add a periodically oscillatingpotential that rises linearly along the lattice,

H (t) = HJ + HU + E cos(!t)∑jxj (nj↑ + nj↓) . (3.19)

Here, E and ! are the amplitude and frequency of the driving, and xj is the spatialcoordinate along the chain. The geometry of the chain is determined by the lattice


constant, which we set as the unit of distance, and the intracell distance b ∈ [0, 1], suchthat xj = ⌊(j − 1)/2⌋ + b(j − 1 mod 2). Since the Hamiltonian is periodic in time wecan apply Floquet theory and obtain a time-independent e�ective Hamiltonian in thehigh-frequency regime, see appendix 3.A. As it turns out, when the leading energyscale is that of the of the interaction between particles, the e�ective Hamiltonian fordoublons is the same as the one in Eq. (3.16) with renormalized hopping amplitudes

J eff1 = 0(2Eb! )

2J 2(1 + �)2U , (3.20)

J eff2 = 0(2E(1 − b)

! )2J 2(1 − �)2

U , (3.21)

where 0 denotes the zeroth order Bessel function of the �rst kind. The factor 2 in theargument of 0 comes from the doublon’s twofold electric charge. We show the e�ectof this renormalization in Fig. 3.2 (bottom row); as the e�ective tunneling reduces inmagnitude the bulk bands become narrower, and two Shockley edge states are pulledfrom the bottom of the lower band. The geometry, which so far did not played any role,now becomes important in this renormalization of the hoppings. The simplest case isfor b = 1/2, in which both hoppings are renormalized by the same factor. We remarkthat the on-site e�ective chemical potential, being a local operator, commutes with theperiodic driving potential, and so it is not renormalized.

This approach has the peculiarity that it only relies on the Δ� produced by theinteraction, and not on the topology of the chain. Thus, it can be used to producethe transfer of doublons in normal chains with an odd number of sites, see Fig. 3.2(bottom row), contrary to the static approach. Notice that the high-frequency e�ectiveHamiltonian preserves space inversion symmetry, which guarantees that the Shockleyedge states hybridize forming even and odd combinations under space inversion, justlike topological edge states do. However, Shockley edge states lack the protectionagainst certain types of disorder, such as o�-diagonal disorder (see section 4.1.1), thattopological edge states have.

Last, we can combine both approaches, restoring the symmetries of the SSH model,and being able to modify the system’s topology with the ac �eld, provided b ≠ 1/2 [93]. InFig. 3.3 we compare the exact quasienergies of the doublon states with the quasienergiesgiven by the e�ective Hamiltonian. Both agree as long as photon-resonance e�ects arenegligible, i.e., E ≲ U (see appendix 3.A). For �eld parameters such that |J eff1 | < |J eff2 |,the system supports a pair of topological edge states, which allow for the transfer ofdoublons.


Figure 3.2: Spectrum of the e�ective Hamiltonian (left), and associated doublon dynam-ics (right), computed numerically solving the Schrödinger equation in the two-particlesinglet subspace. In all the cases shown, a doublon is initially on the 1st site of the lattice.Case without any external in�uence (top row). The spectrum shows the absence ofedge states for any value of � . The parameters for the dynamics are: � = −0.3, U = 10Jand N = 5. Case with a compensating local potential Vgate on the ending sites of thechain (middle row). A pair of topological edge states appears for negative values of� . The parameters for the dynamics are the same as in the previous case. Case withan external ac-�eld with parameters: E = 3.2J , ! = 2J and b = 1/2 (bottom row). Thespectrum shows a pair of Shockley edge states that separate from the bottom of thelower band. The dynamics is for a chain with 11 sites, � = 0, and U = 16J .


0:0 2:5 5:0 7:5 10:0 12:5 15:0 17:5 20:0

2E=!

0:0

0:1

0:2

0:3

0:4

0:5

Qua

siene

rgie

sŒJ�

Figure 3.3: Quasienergy spectrum for a chain with a compensating local potential Vgateand an external ac �eld, as a function of the ac �eld amplitude. The parameters are� = 0.1, N = 7, U = 16J , b = 0.6, and ! = 2J . Exact quasienergies (dots) have beenobtained diagonalizing the time evolution operator for one period. We only showthe quasienergies corresponding to the 2N Floquet modes with largest total doubleoccupancy. The lines correspond to the approximate quasienergies given by the e�ectiveHamiltonian. Shaded areas mark the regions where the system is in the non-trivialphase, according to the values of |J eff1 | and |J eff2 |.


3.2.2. Dynamics in the 3 and Lieb latticesWe will now analyze the consequences of the e�ective local chemical potential on thedoublon dynamics in 2D lattices. We will consider lattices with homogeneous hoppingamplitudes threaded by a static magnetic �ux in the presence of a circularly polarizedac �eld. They are modelled by the Hamiltonian

H (t) = −J ∑⟨i,j⟩, �

ei�ijc†i�cj� + U ∑jnj↑nj↓ +∑

j, �Vj(t)nj� , (3.22)

with Vj(t) = xjE cos(!t) + yjE sin(!t), where (xj , yj) ≡ rj are the coordinates of site j.The sum in the hopping term runs over each oriented pair of nearest-neighbor sites.The magnetic �ux induces complex phases in the hoppings such that the sum of thephases around a closed loop equals 2�Φ/Φ0, where Φ is the total �ux threading the loopand Φ0 is the magnetic �ux quantum.

The e�ective Hamiltonian for doublons generalizes in a straightforward mannerfor 2D lattices (see appendix 3.A),

Heff = Jeff ∑⟨i,j⟩

ei2�ijd†i dj +∑j�jd†j dj , (3.23)

with e�ective hopping and chemical potentials

Jeff =2J 2U 0(

2Ea! ) , �j =

2J 2U zj , (3.24)

where zj is the coordination number (the number of nearest neighbors) of site j. Inthis e�ective Hamiltonian we have neglected again interaction terms, since we areconsidering just one doublon. Importantly, the hopping renormalization is isotropicbecause the ac �eld polarization is circular and all neighboring sites are the samedistance apart; |ri − rj | = a for all neighboring i and j.

As we can see, the ac driving allows us to independently tune the e�ective hoppingamplitude with respect to the e�ective local chemical potential. This has a big impact onthe dynamics of doublons in lattices that can be divided into sublattices with di�erentcoordination numbers, such as the Lieb lattice, and the 3 lattice, shown in Fig. 3.4. Ascan be seen in the dynamics, the doublon moves mostly through sites with the samecoordination number, an e�ect we have termed sublattice dynamics.

To understand why, it is useful to look at the e�ective Hamiltonian in momentumrepresentation, which in the absence of an external magnetic �ux adopts the same formfor both lattices Heff = ∑k V †

k HkVk, with

Hk =⎛⎜⎜⎝

Δ� f1(k) f2(k)f ∗1 (k) 0 0f ∗2 (k) 0 0

⎞⎟⎟⎠, Vk =

⎛⎜⎜⎝

dAkdB1kdB2k

⎞⎟⎟⎠. (3.25)


Figure 3.4: Doublon dynamics on a �nite piece of a Lieb lattice (a) and a 3 lattice (b).In both cases the doublon is initially on the blue dot. Parameters: Φ = 0, U = 16J ,! = 2J , and E = 4.8J (a), E = 4.2J (b). The grey line is the sum of the double occupancyon the four colored sites. In the Lieb lattice, the double occupancy on the green andyellow sites is the same. The dynamics have been obtained solving numerically thetime-dependent Schrödinger equation in the two-particle singlet subspace.

Here, d�k is the annihilation operator of a doublon with momentum k in sublattice� ∈ {A, B1, B2}. The eigenenergies and eigenstates are as follows:

�0k = 0 , �±k =12 [Δ� ±

√4|f1(k)|2 + 4|f2(k)|2 + Δ�2] , (3.26)

||u0k⟩ ∝ [−f2(k)f1(k)

d†B1k + d†B2k] |0⟩ , (3.27)

||u±k⟩ ∝ [�±kf ∗2 (k)

d†Ak +f ∗1 (k)f ∗2 (k)

d†B1k + d†B2k] |0⟩ . (3.28)

Note how the states of the �at band do not have weight on the A sites of the lattice.The chemical potential di�erence Δ� = 2J 2(zA − zB)/U produces a splitting between theupper band and the rest of the bands, see Fig. 3.5. The functions f1 and f2 depend onthe particular lattice geometry as shown in the table below,


Lattice f1(k) f2(k)

3 Jeff [e−i(kx+

ky√3)

12 + ei(kx−

ky√3)

12 + ei

ky√3 ] f2(k) = f ∗1 (k)

Lieb 2Jeff cos ( kx2 ) 2Jeff cos(ky2 )

They are proportional to Jeff , which can be tuned by the ac driving. In particular, therelative weight on the A sublattice of the Bloch states corresponding to the upper(lower) band can be increased (reduced) by tuning the ac �eld parameters closer to azero of the Bessel function.

Figure 3.5: Doublon energy bands for the Lieb lattice. The e�ective local potentialopens a gap between the upper band and the rest. The driving allows the band widthto be reduced, �attening the bands, while keeping the gap the same.

When studying quantum walks [94], i.e., the coherent evolution of particles innetworks, it is natural to ask about the probability of �nding a particle that was initiallyon site i to be on site j after a certain time t , that is, pij(t) ≡ |⟨i|U (t)|j⟩|2 = |⟨i|e−iH t |j⟩|2.Using (3.23) as the e�ective single-particle Hamiltonian for the doublon, we de�nepA(t) ≡ ∑i, j∈A pij(t)/NA, which is the probability for the doublon to remain in sublatticeA at time t ; NA is the total number of sites that belong to sublattice A. To demonstratesublattice con�nement, we can compute the long-time average

pA ≡ limt→∞

1t ∫

t

0dt′ pA(t′) . (3.29)

According to the de�nition, the probability pA(t) is ‖UA(t)‖2/NA, where ‖⋅‖ denotes theHilbert-Schmidt norm, and UA(t) = PAU (t)PA is the time evolution operator projected


on the subspace of the A sublattice. Using the spectral decomposition

U (t) =∑k

∑�=0,±

e−i��k t ||u�k⟩⟨u�k || , (3.30)

we can express

‖UA(t)‖2 =∑k

|||||e−i�+k t

1 + g+(k)+ e−i�−k t1 + g−(k)

|||||

2

(3.31)

= ∑k, �=±

1[1 + g� (k)]2

+∑k

2 cos (�+k t − �−k t)[1 + g+(k)] [1 + g−(k)]

, (3.32)

where we have de�ned g±(k) = [|f1(k)|2 + |f2(k)|2] (�±k)−2. The time average is, thus,

given by

pA =1V ∫

FBZd2k

{ 1[1 + g+(k)]2

+ 1[1 + g−(k)]2

}. (3.33)

Here, we have taken the thermodynamic limit, replacing the sum over momenta by anintegral on the �rst Brillouin zone (FBZ); V stands for the FBZ area. As can be seen inFig. 3.6(a) the probability pA can be enhanced by tuning the ratio Δ�/Jeff to larger values,meaning that it is possible to con�ne the doublon’s dynamics to a single sublattice bysuitably changing the ac �eld parameters. We have also computed the dependence ofpA with the magnetic �ux threading the smallest plaquette (the smallest closed path inthe lattice), see Fig. 3.6(b); however, its variation turns out to be minor with pA gentlyincreasing as the �ux is tuned away from 2Φ/Φ0 = 1/2. A much stronger dependence isobserved for the 3 than for the Lieb lattice. This is to be expected as Aharonov-Bohmphases have more dramatic e�ects in the 3 lattice, notably the caging e�ect that occursfor a magnetic �ux Φ/Φ0 = 1/2 in the singly-charged particle case [95].

It is worth mentioning that a similar e�ect constrains the motion of doublons inany lattice with boundaries. The sites on the edges necessarily have fewer neighborsthan those in the bulk and therefore have a smaller chemical potential. This produceseigenstates localized on the edges, which are of the Shockley or Tamm type. As aconsequence, the doublon’s dynamics can be con�ned to just the edges of the lattice.Furthermore, since having non-trivial topology in 2D does not require any symmetry,in the presence of a nonzero magnetic �ux any lattice can support both Shockley-likeedge states and topological chiral edge states at the same time. Let us be more speci�c.When comparing the e�ective model (3.23) with that of a Chern insulator [7], the onlydi�erence is the local chemical potential term. It is well known that strong disorderpotentials eventually destroy the topological properties of Chern insulators as theytransition to a trivial Anderson insulator by a mechanism known as “levitation andannihilation” of extended states [96, 97]. Nonetheless, the chemical potential term in


Figure 3.6: Probability to remain in sublattice A as a function of the ratio Δ�/Jeff , forΦ = 0 (a), and as a function of the magnetic �ux per plaquette, for Δ�/Jeff = 3 (b).

our Hamiltonian constitutes a very particular form of disorder that does not a�ectthe topology of the system. In Fig. 3.7 we show the energy spectrum of a ribbon ofthe Lieb lattice in the presence of a magnetic �ux. There, we can observe topologicaledge states appearing in the minigaps opened by the magnetic �ux that propagate in a�xed direction depending on their energy and the edge where they localize. We canalso �nd Shockley-like edge states that can propagate in both directions along eachedge. When reducing the e�ective hopping, these states are pulled further out of thebulk minibands, making them interfere less with the topological edge states. After thisanalysis we conclude that, depending on their energy, a doublon can propagate chirallyor not along the edges of a lattice threaded by a magnetic �ux.


Figure 3.7: Energy spectrum of a �nite ribbon of the Lieb lattice, with Ny = 50 unitcells along the y direction in the presence of a magnetic �ux Φ/Φ0 = 1/10. The ac �eldin the case shown in the right plot is such that 0 (2Ea/!) = 1/2. The color indicatesthe localization of the state along the �nite dimension of the lattice: green and purpledenote states localized on each edge of the ribbon (topological and Shockley-like edgestates), while blue is used for bulk states that are not localized (minibands). Topologicaledge states always connect di�erent minibands, while Shockley edge states do notnecessarily do so.


3.3. Doublon decay in dissipative systemsThe high controllability and isolation achieved in cold atom experiments make thema great platform for observing doublons. But can doublons be observed in other kindof systems? Nowadays, solid-state devices such as quantum dot arrays are beinginvestigated as platforms for quantum simulation. However, phonons, nuclear spins,and �uctuating charges and currents make for a much noisier environment in thesesetups as compared with others [69]. In this section we investigate how the coupling tothe environment may a�ect the stability of doublons in QD arrays and give an estimateof their lifetime in current devices.

We will analyze the case of a 1D array of N quantum dots, see Fig. 3.8. Electronstrapped in the QD array are modelled by the Hubbard Hamiltonian with an homoge-neous hopping amplitude J and interaction strength U . For the environment, we assumethe chain is coupled to several independent baths of harmonic oscillators. The systemand the environment can be modelled as a whole by the Hamiltonian H = HS +HB +HI ,with

HS = −J ∑j, �

(c†j+1�cj� + H.c.) + U ∑jnj↑nj↓ , (3.34)

HB =∑j, n!na†njanj , (3.35)

HI =∑jXjBj , Bj =∑

ngn(a†nj + anj) . (3.36)

Here, Bj is the collective coordinate of the jth bath that couples to the system operatorXj , which will be speci�ed below. Moreover, we assume that all baths are equal andstatistically independent.

In the Markovian regime, the time evolution of the system’s density matrix �, canbe suitably described by a Bloch-Red�eld master equation of the form [64, 65] (seeappendix 3.B)

� = −i[HS , �] −∑j[Xj , [Qj , �]] −∑

j[Xj , {Rj , �}]

≡ −i[HS , �] + [�] , (3.37)

with the anticommutator {A, B} = AB + BA and

Qj =1� ∫

∞

0d� ∫

∞

0d! (!)Xj(−� ) cos!� , (3.38)

Rj =−i� ∫

∞

0d� ∫

∞

0d! (!)Xj(−� ) sin!� . (3.39)

3.3. DOUBLON DECAY IN DISSIPATIVE SYSTEMS 41

Current noise

Charge noise

Figure 3.8: Schematic picture of the system under consideration. A QD array modelledas a tight-binding 1D lattice is occupied by two electrons. An initial state with a doublyoccupied site (doublon) may decay dissipatively into a single-occupancy state withlower energy. The released energy is of the order of the on-site interaction U and willbe absorbed by the heat baths representing environmental charge and current noise.

The tilde denotes the interaction picture, Xj(−� ) = e−iHS�XjeiHS� , while the spectraldensity of the baths is (!) = � ∑n |gn|2�(! − !n), and (!) = (!) coth(�!/2) is theFourier transformed of the symmetrically ordered equilibrium autocorrelation function⟨{Bj(� ), Bj(0)}⟩/2. (!) and (!) are independent of the bath subindex j since all bathsare identical. We will assume an ohmic spectral density (!) = ��!/2, where thedimensionless parameter � characterizes the dissipation strength.

3.3.1. Charge noiseFluctuations of the background charges in the substrate essentially act upon the chargedistribution of the chain. We model it by coupling the occupation of each site to a heatbath, such that

HI =∑j, �nj�Bj , Xj = nj↑ + nj↓ . (3.40)

This fully speci�es the master equation (3.37).To get a qualitative understanding of the decay dynamics of a doublon, let us start

by discussing the time evolution of the total double occupancy

D ≡ ∑jnj↑nj↓ =∑

jd†j dj , (3.41)

for an initial doublon state, shown in Fig. 3.9(a). For � = 0, i.e., in the absence ofdissipation, the two electrons will essentially remain together throughout time evolution.


However, since the doublon states are not eigenstates of the system Hamiltonian, weobserve some slight oscillations of the double occupancy. Still the time average of thisquantity stays close to unity. On the contrary, if the system is coupled to a bath, doublonswill be able to split, releasing energy into the environment. Then the density operatoreventually becomes the thermal state �∞ ∝ e−�HS . Depending on the temperature andthe interaction strength, the corresponding asymptotic doublon occupancy ⟨D⟩∞ maystill assume an appreciable value.

To gain a quantitative insight, we decompose our master equation (3.37) into thesystem eigenbasis and obtain a form convenient for numerical treatment (appendix 3.B).A typical time evolution of the total double occupancy exhibits an almost monoex-ponential decay, such that the doublon life time T1 can be de�ned as the time when

⟨D⟩T1 − ⟨D⟩∞1 − ⟨D⟩∞

= 1e . (3.42)

The corresponding decay rate Γ = 1/T1 is shown in Fig. 3.10 as a function of thetemperature and interaction strength for a �xed small � .

0:0 2:5 5:0 7:5 10:0

time�J�1

�0:4

0:6

0:8

1:0

hDi

(a)

˛ D 0˛ D 0:04

0:0 0:5 1:0 1:5 2:0

time�J�1

�0:7

0:8

0:9

1:0

hP1i

(b)

kBT D 204J

0:01J

816J

Figure 3.9: Time evolution of the double occupancy in a system with charge noise. Theinitial state consists of a doublon localized in a particular site of a chain with periodicboundary conditions. Parameters: N = 5, U = 10J and � = 0.04. (a) Comparisonbetween free dynamics (� = 0) and dissipative dynamics (� ≠ 0). Temperature is set tokBT = 0.01J . The green line corresponds to the occupancy of the high-energy subspacefor the case with � ≠ 0 and illustrates the bound given in Eq. (3.44). (b) Decay ofthe high-energy subspace occupancy for di�erent temperatures ranging from 0.01J to1000J . The slope of the curves at time t = 0 is the same in all cases and coincides withthe value given by Eq. (3.47).

An analytical estimate for the decay rates can often be gained from the behaviorat the initial time t = 0, i.e. from �0. In the present case, however, the calculation is


hindered by the fast initial oscillations witnessed in Fig. 3.9(a). To circumvent thisproblem, we focus instead on the occupancy of the high-energy subspace, ⟨P1⟩, withP1 being the projector onto the high energy subspace, shown in Fig. 3.9(b). Using theSchrie�er-Wol� transformation derived in section 3.1 we can express it in terms of theprojector onto the doublon subspace, PD , as

P1 = PD +1U (H +

J + H −J ) + O (U −2) . (3.43)

It turns out that this quantity evolves more smoothly while it decays also on the timescale T1. To understand this similarity, notice that

|tr (P1�) − tr (D�)| ≤√2‖�‖

√N − tr (P1PD) ≃

2√2NJU , (3.44)

where we have used the Cauchy-Schwarz inequality for the inner product of operators,(A, B) = tr (A†B), and the perturbative expansion of P1 mentioned before. The reasonfor its lack of fast oscillations is that the projector P1 commutes with the systemHamiltonian, so that its expectation value is determined solely by dissipation.

Following our hypothesis of a monoexponential decay, we expect

⟨P1⟩ ≃ Δe−Γt + ⟨P1⟩∞ , (3.45)

therefore

Γ ≃ − 1Δd⟨P1⟩dt

||||t=0= − tr (P1[�0])

⟨P1⟩0 − ⟨P1⟩∞. (3.46)

This expression still depends slightly on the speci�c choice of the initial doublon state.To obtain a more global picture, we consider an average over all doublon states, whichcan be performed analytically [98] (see appendix 3.C). Substituting the expression forthe Liouvillian in Eq. (3.46), we �nd the average decay rate

Γ = 1NΔ∑

jtr (PD[Qj , [Xj , P1]]) − tr (PD{Rj , [Xj , P1]}) . (3.47)

For a further simpli�cation, we have to evaluate the expressions (3.38) and (3.39) whichis possible by approximating the interaction picture coupling operator as Xj(−� ) ≃Xj − i� [HS , Xj]. This is justi�ed as long as the decay of the environmental excitations ismuch faster than the typical system evolution, i.e., in the high-temperature regime (HT).Inserting our approximation for Xj and neglecting the imaginary part of the integrals,we arrive at

Qj ≃12 lim!→0+

(!)Xj =�2 �kBTXj , (3.48)

Rj ≃ −12 lim!→0+

′(!)[HS , Xj] =�4 �[HS , Xj] . (3.49)


With these expressions, Eq. (3.47) results in a temperature independent decay rate.Notice that any temperature dependence stems from the Qj in the �rst term of Eq. (3.47),which vanishes in the present case. While this observation agrees with the numerical�ndings in Fig. 3.9(b) for very short times, it does not re�ect the temperature dependentdecay of ⟨P1⟩ at the more relevant intermediate stage.

This particular behavior hints at the mechanism of the bath-induced doublon decay.Let us remark that for the coupling to charge noise, Xj commutes with D. Therefore,the initial state is robust against the in�uence of the bath. Only after mixing with thesingle-occupancy states due to the coherent dynamics, the system is no longer in aneigenstate of Xj , such that decoherence and dissipation become active. Thus, it is thecombined action of the system’s unitary evolution and the e�ect of the environmentwhich leads to the doublon decay. An improved estimate of the decay rate, can becalculated by averaging the transition rate of states from the high-energy subspaceto the low-energy subspace. Let us �rst focus on regime kBT ≳ U in which we canevaluate the operators Qj in the high-temperature limit. Then the average rate can becomputed using expression (3.47) replacing PD by P1. With the perturbative expansionof P1 in Eq. (3.43) we obtain to leading order in J /U the averaged rate

ΓHT ≃4��J 2U 2Δ (2kBT + U ) , (3.50)

valid for periodic boundary conditions. For open boundary conditions, the rate acquiresan additional factor (N − 1)/N . Notice that we have neglected back transitions viathermal excitations from singly occupied states to doublon states. We will see that thisleads to some deviations when the temperature becomes extremely large. Nevertheless,we refer to this case as the high-temperature limit.

In the opposite limit, for temperatures kBT < U , the decay rate saturates at a constantvalue. To evaluate Γ in this limit, it would be necessary to �nd an expression for Xj(−� )dealing properly with the � -dependence for evaluating the noise kernel, a formidabletask that may lead to rather involved expressions. However, one can make someprogress by considering the transition of one initial doublon to one particular single-occupancy state. This corresponds to approximating our two-particle lattice model bya dissipative two-level system for which the decay rates in the Ohmic case can be takenfrom the literature [99, 100] (see appendix 3.D). Relating J to the tunnel matrix elementof the two-level system and U to the detuning, we obtain the temperature-independentexpression

ΓLT ≃8��J 2UΔ , (3.51)

which formally corresponds to Eq. (3.50) with the temperature set to kBT = U /2.


Figure 3.10: Comparison between the numerically computed decay rate and the analyticformulas (3.50) and (3.51) for a chain with N = 5 sites and periodic boundary conditionsin the case of charge noise. The dissipation strength is � = 0.02. (a) Dependence onthe interaction strength for a �xed temperature kBT = 20J . (b) Dependence on thetemperature for a �xed interaction strength U = 20J .

3.3.2. Current noiseFluctuating background currents mainly couple to the tunnel matrix elements of thesystem. Then the system-bath interaction is given by

HI =∑j, �

(c†j+1�cj� + c†j�cj+1�)Bj , (3.52)

Xj = c†j+1�cj� + c†j�cj+1� . (3.53)

Depending on the boundary conditions, the sum may include the term connecting the�rst and last QD of the array. The main di�erence with respect to the case of chargenoise is that now HI does not commute with the projector onto the doublon subspaceand, thus, generally tr (D[�0]) ≠ 0. This allows doublons to decay without havingto mix with single-occupancy states. Therefore, for the same value of the dissipationstrength � , the decay may be much faster.

As in the last section, we proceed by calculating analytical estimates for the decayrates. However, the time evolution is no longer monoexponential. In this case, weestimate the rate from the slope of the occupancy ⟨P1⟩ at initial time,

Γ ≃ − d⟨P1⟩dt

||||t=0= − tr (P1[�0]) . (3.54)

We again perform the average over all doublon states for �0 in the limits of high andlow temperatures. For periodic boundary conditions, we obtain to lowest order in J /U


Figure 3.11: Numerically obtained decay rate in comparison with the approxima-tions (3.54), (3.55) and (3.56) for a chain with N = 5 sites and periodic boundaryconditions in the case of current noise with strength � = 0.02. The results are plottedas a function of (a) the interaction and the temperature kBT = 20J and (b) for a �xedinteraction U = 20J as a function of the temperature.

the high and low temperature rates

ΓHT = 2�� (2kBT + U ) , (3.55)ΓLT = 4��U , (3.56)

respectively, while open boundary conditions lead to the same expressions but with acorrection factor (N − 1)/N . In Fig. 3.11, we compare these results with the numericallyevaluated ones as a function of the interaction and the temperature. Both show thatthe analytical approach correctly predicts the (almost) linear behavior at large valuesof U and kBT , as well as the saturation for small values. However, the approximationslightly overestimates the in�uence of the bath.

While the rates re�ect the decay at short times, it is worthwhile to comment onthe long time behavior under the in�uence of current noise. As it turns out, thesteady state is not unique. The reason for this is the existence of a doublon state|Φ⟩ = 1√

N ∑Nj=1(−1)jc†j↑c†j↓|0⟩ which is an eigenstate of HS without any admixture of single-

occupancy states. Since Xj |Φ⟩ = 0 for all j, current noise may a�ect the phase of |Φ⟩, butcannot induce its dissipative decay. For a closed chain with an odd number of sites, bycontrast, the alternating phase of the coe�cients of |Φ⟩ is incompatible with periodicboundary conditions, unless a �ux threads the ring. As a consequence, the state of thechain eventually becomes the thermal state �∞ ∝ e−�HS . The di�erence is manifest inthe �nal value of the doublon occupancy at low temperatures. For closed chains withan odd number of sites, it will fully decay, while in the other cases, the population of|Φ⟩ will survive.


3.3.3. Experimental implicationsA current experimental trend is the fabrication of larger QD arrays [39, 101], whichtriggered our question on the feasibility of doublon experiments in solid-state systems.While the size of these arrays would be su�cient for this purpose, their dissipativeparameters are not yet fully known. For an estimate we therefore consider the valuesfor GaAs/InGaAs quantum dots which have been determined recently via Landau-Zenerinterference [69]. Notice, that for the strength of the current noise, only an upper boundhas been reported. We nevertheless use this value, but keep in mind that it leads to aconservative estimate. In contrast to the former sections, we now compute the decayfor the simultaneous action of charge noise and current noise.

Figure 3.12 shows the T1 times for two di�erent interaction strengths. It reveals thatfor low temperatures T ≲ J /kBT , the life time is essentially constant, while for largertemperatures, it decreases moderately until kBT comes close to the interaction U . Forhigher temperatures, Γ starts to grow linearly. On a quantitative level, we expect lifetimes of the order T1 ∼ 5 ns already for moderately low temperatures T ≲ 100mK. Sincewe employed the value of the upper bound for the current noise, the life time might beeven larger.

Considering the estimates for the decay rates at low temperatures, Eqs. (3.51) and(3.56), separately, we conclude that for smaller values of U , current noise becomes lessimportant, while the impact of charge noise grows. Therefore, a strategy for reachinglarger T1 times is to design QD arrays with smaller on-site interaction, such that theratio U /J becomes more favorable. The largest T1 is expected in the case in whichboth low-temperature decay rates are equal, ΓchargeLT = ΓcurrentLT , which for the presentexperimental parameters is found at U ∼ 10J . This implies that in an optimized device,the doublon life times could be larger by one order of magnitude to reach values ofT1 ∼ 50 ns, which is corroborated by the data in the inset of Fig. 3.12.


10�2 10�1 100 101

T [K]

100

T1

[ns]

U D 2:6 meV4:16 meV

10�2 100101

102

130 �eV

Figure 3.12: Doublon life time as a function of the temperature for di�erent interactionstrengths. The parameters are: �charge = 3 × 10−4, �current = 5 × 10−6, and J = 13 �eV. TheT1 time has been measured for a doublon starting in the middle of a chain with N = 5and open boundary conditions. Inset: T1 time for the optimized value of the interaction,U = 10J = 130 �eV and a current noise with �current = 2 × 10−6.

3.4. SummaryFor understanding the dynamics of doublons, we have derived an e�ective single-particle Hamiltonian taking into account also the e�ect of a periodic driving on thelattice. It contains two terms: one corresponding to an e�ective doublon hoppingrenormalized by the driving, and another one corresponding to an e�ective on-sitechemical potential, with the peculiarity of being proportional to the number of neighborsof each site. Importantly, in the regime where the Hubbard interaction is larger thanthe frequency of the driving and these are both larger than any other energy scaleof the system, the driving allows to tune the doublon hopping independently of thee�ective chemical potential. This produces several interesting phenomena regardingthe doublons’ motion:

(1) In any �nite lattice doublons experience an e�ective chemical potential di�erencebetween the sites on the edges and the rest of the sites. This allows the generationof Shockley-like edge states by reducing the doublon hopping with the driving.

(2) In the SSH-Hubbard model, the chemical potential di�erence between the endingsites of the chain and the rest of the sites causes the disappearance of the topolog-ical edge states for doublons. This chemical potential di�erence breaks the chiralsymmetry of the SSH model, which is essential for having non-trivial topology.On the other hand, for 2D lattices threaded by a magnetic �ux, no symmetry is

3.4. SUMMARY 49

required for having non-trivial topology, thus, they do support topological edgestates for doublons, which may coexist with Shockley-like edge states inducedby the driving.

(3) Both topological and Shockley-like edge states allow for the direct long-rangetransfer of doublons between distant sites in the edges of a �nite lattice.

(4) In lattices with sites with di�erent coordination numbers it is possible to con�nethe doublon’s motion to a single sublattice at the expense of slowing it down byreducing the doublon hopping with the driving.

We have also studied the stability of doublons in quantum dot arrays in the presence ofcharge noise and current noise. While the dependence on temperature of the doublon’slifetime T1 is similar for both types of noise, the dependence with the interactionstrength U is very di�erent. For charge noise T1 ∼ U , whereas for current noiseT1 ∼ U −1, in the low temperature regime (kBT < U ). In current devices we predict adoublon lifetime of the order of 10 ns, although it can be improved up to one order ofmagnitude in devices speci�cally designed to that end.

Appendices

3.A. Effective Hamiltonian for doublonsWe start from a Fermi-Hubbard model with an ac �eld that couples to the particledensity and a magnetic �ux that introduces complex phases in the hoppings. TheHamiltonian of the system is H (t) = HJ + HU + HAC(t), with

HAC(t) =∑jVj(t)(nj,↑ + nj,↓) . (3.57)

For a time-periodic Hamiltonian, H (t + T ) = H (t), with T = 2�/!, Floquet’s theoremallows us to write the time-evolution operator as

U (t2, t1) = e−iK (t2)e−iHeff (t2−t1)eiK (t1) , (3.58)

where Heff is a time-independent, e�ective Hamiltonian, and K (t) is a T -periodic Her-mitian operator. Heff governs the long-term dynamics, whereas e−iK (t), also known asthe micromotion-operator, accounts for the fast dynamics occurring within a period.Following several perturbative methods [60, 61], it is possible to �nd expressions forthese operators as power series in 1/!,

Heff =∞∑n=0

H (n)eff!n , K (t) =

∞∑n=0

K (n)(t)!n . (3.59)

These are known in the literature as high-frequency expansions (HFE). The di�erentterms in these expansions have a progressively more complicated dependence on theFourier components of the original Hamiltonian,

Hq =1T ∫

T

0dt H (t)ei!qt . (3.60)

The �rst three of them are:

H (0)eff = H0 , (3.61)

H (1)eff =∑

q≠0

H−qHqq , (3.62)

H (2)eff = ∑

q, p≠0(H−qHq−pHp

qp − H−qHqH0q2 ) . (3.63)

Before deriving the e�ective Hamiltonian, it is convenient to transform the originalHamiltonian into the rotating frame with respect to both the interaction and the ac


�eld [102],

H (t) = †(t)H (t) (t) − i †(t))t (t) , (3.64)

(t) = exp [−iHU t − i ∫ dt HAC(t)] . (3.65)

Noting that for fermions

eiHU tc†i�cj�e−iHU t = [1 − ni� (1 − eiU t)] c†i�cj� [1 − nj� (1 − e−iU t)] , (3.66)

the Hamiltonian in the rotating frame can be written as

H (t) = − ∑i, j, �

Jij(t) (T 0ij� + eiU tT +

ij� + e−iU tT −ij�) , (3.67)

with Jij(t) = JijeiA(t)⋅dij . Note that this is a di�erent way of expressing the coupling to anelectric �eld described by the vector potential A(t). In the case of circular polarization,A(t) = (sin(!t), − cos(!t)) E/!; dij = ri − rj is the vector connecting sites i and j. Wehave de�ned:

T 0ij� = ni�c†i�cj�nj� + ℎi�c†i�cj�ℎj� , (3.68)T +ij� = ni�c†i�cj�ℎj� , T −

ij� = (T +ji� )† = ℎi�c†i�cj�nj� . (3.69)

The operators T 0ij� involve hopping processes that conserve the total double occupancy,

while T +ij� and T −

ij� raise and lower the total double occupancy respectively.In order to apply the HFE method we need to �nd a common frequency. We

will consider �rst the resonant regime, U = l!, and then, by means of analyticalcontinuation, obtain the strongly-interacting limit (U ≫ ! > J ) and the ultrahigh-frequency limit (! ≫ U > J ). The Fourier components of H (t) are

Hq = − ∑i, j, �

(Jij,qT 0ij� + Jij,q+lT +

ij� + Jij,q−lT −ij�) , (3.70)

with

Jij,q =JijT ∫

T

0dt ei E! dxij sin(!t)e−i E! dyij cos(!t)eiq!t (3.71)

= JijT ∫

T

0dt ∑

m, nm (

Edxij! )−n (

Edyij! ) i

nei(m+n+q)!t (3.72)

= Jij ∑nn+q (

Edxij! )n (

Edyij! ) e

−in�/2 (3.73)

= Jije−iq�ijq (E|dij |! ) , (3.74)

3.A. EFFECTIVE HAMILTONIAN FOR DOUBLONS 53

where �ij = arg (dxij + idyij), and q stands for the Bessel function of �rst kind of order q.To go from the �rst to the second line we have used the Jacobi–Anger expansion (thesums run over all positive and negative integers), and we have used Graf’s additiontheorem to derive the last expression. Note that Jij,q = J ∗ji,−q.

Now, the zeroth-order approximation in the HFE is given by:

H (0)eff = −J ∑

i, j, �Jij [0 (

E|dij |! ) T

0ij� + e−il�ijl (

E|dij |! ) T

+ij� + eil�ij−l (

E|dij |! ) T

−ij�] . (3.75)

In contrast to the undriven case, the total double occupancy is not necessarily anapproximate conserved quantity in the regime U ≫ J . There are terms proportional tol (E|dij |/!) that correspond to the formation and dissociation of doublons assisted bythe ac �eld (l-photon resonance). However, for low driving amplitudes (E|dij |/! < l) theprobability for these processes to occur is very small and we can neglect them. It is inthis low amplitude regime where it makes sense to consider an e�ective Hamiltonianfor doublons. We neglect the terms that go with T 0

ij� because they act non-trivially onlyon states with some single-occupancy.

In the next order of the HFE, there are more terms that do not conserve the totaldouble occupancy, which we neglect, and from those which do conserve it, we onlykeep the ones that act non-trivially on the doublon’s subspace:

H (1)eff!

∗= ∑i, j, �

⎡⎢⎢⎢⎣

∑q≠0

2−q+l (

E|dij |! ) |Jij |2T +

ij�T −ji�

q!

+ ∑q≠0

−q+l (E|dij |! )q−l (

E|dij |! ) J 2ijT +

ij�T −ij�

q!

⎤⎥⎥⎥⎦

. (3.76)

Here, the �rst term is equal to

|Jij |2 ∑p≠−l

2p (

E|dij |! )

(l − p)! T +ij�T −

ji� =|Jij |2U ∑

p≠−l

2p (

E|dij |! )

1 − p!/U (ni�ni� − ni�ni�nj�nj�) , (3.77)

and the second term is equal to

J 2ij ∑p≠−l

p (E|dij |! )−p (

E|dij |! )

(l − p)! T +ij�T −

ij� =

J 2ijU ∑

p≠−l

p (E|dij |! )−p (

E|dij |! )

1 − p!/U c†i�c†i�cj�cj� . (3.78)


In the limit U ≫ ! > J , p!/U ≪ 1 and we can approximate all the denominators inthe above expressions as 1. Note that for �xed argument � , the Bessel functions q(�)decay for increasing order |q|. Also, when analytically continuing the formulas forvalues of U other than multiples of !, we may safely neglect the restriction p ≠ −l.Finally, using the identities ∑q 2

q (�) = 1 and ∑q q(�)k−q(�) = k(� + �), we arrive at

HU≫!eff =∑

i, jJ effij d†i dj +∑


i, j

2|Jij |2U d†i did†j dj , (3.79)

J effij ≡ 2J 2ijU 0 (

2E|dij |! ) , �i ≡ ∑

j

2|Jij |2U . (3.80)

For completeness we give also the result in the other limit: ! ≫ U > J . Now p!/Uis very large and all the terms in the sums are very small except those for p = 0. Thee�ective Hamiltonian in this case would be:

H!≫Ueff =∑

i, jJ effij d†i dj +∑


i, j|J effij |d†i did†j dj , (3.81)

J effij ≡ 2J 2ijU 2

0 (E|dij |! ) , �i ≡ ∑

j|J effij | . (3.82)

It is worth mentioning that these results could also be obtained by applying the HFEsequentially, integrating �rst the fast varying terms corresponding to the leading energyscale in the system. We also note that higher order corrections will include complexnext-nearest-neighbor hoppings that break the time-reversal symmetry, even in systemswithout any external magnetic �ux.

3.B. Bloch-Redfield master eqationExpanding the integrand of Eq. (2.19) we get

trB[HI , [HI (−� ), �S ⊗ �B]] =∑j, k

[Cjk(� )XjXk(−� )�S − Ckj(−� )Xj�SXk(−� )

− Cjk(� )Xk(−� )�SXj + Ckj(−� )�SXk(−� )Xj] . (3.83)

Here, we have de�ned ⟨Bj(t)Bk(t′)⟩ ≡ Cjk(t − t′). If bath operators are independent fromeach other Cjk(� ) = Cj(� )�jk . Then, splitting the correlation functions into symmetricand antisymmetric parts Cj(� ) = Sj(� ) + Aj(� ),

Sj(� ) =Cj(� ) + Cj(−� )

2 , Aj(� ) =Cj(� ) − Cj(−� )

2 , (3.84)

3.B. BLOCH-REDFIELD MASTER EQUATION 55

Eq. (3.83) can be rewritten as

trB[HI , [HI (−� ), �S ⊗ �B]] =∑jSj(� )[Xj , [Xj(−� ), �S]] +∑

jAj(� )[Xj , {Xj(−� ), �S}] . (3.85)

Putting everything together we get

� = −i[HS , �] −∑j[Xj , [Qj , �]] −∑

j[Xj , {Rj , �}] , (3.86)

where we have dropped the subindex “S” of the system’s reduced density matrix, andwe have de�ned

Qj ≡ ∫∞

0d� Sj(� )Xj(−� ) , Rj ≡ ∫

∞

0d� Aj(� )Xj(−� ) . (3.87)

So far the derivation remained rather general, we will now particularize to thecase where the environment is composed of several independent baths of harmonicoscillators HB = ∑j,n !na†njanj , that couple to the system via Bj = ∑n gnanj + H.c. Weconsider that these baths are identical and statistically independent. They are allcharacterized by the same spectral density (!) ≡ � ∑n |gn|2�(! − !n), which describeshow the system couples to the di�erent modes of a bath. Both S(� ) and A(� ) can beexpressed in terms of this spectral density as

C(� ) = 1� ∫

∞

0d! (!)

{ei!�nB(!) + e−i!� [1 + nB(!)]

}, (3.88)

S(� ) = 1� ∫

∞

0d! (!) coth(�!/2) cos!� , (3.89)

A(� ) = −i� ∫∞

0d! (!) sin!� . (3.90)

Here, nB(!) = (e�! − 1)−1 is the bosonic thermal ocupation number. Thus,

Qj =1� ∫

∞

0d� ∫

∞

0d! (!)Xj(−� ) cos!� , (3.91)

Rj =−i� ∫

∞

0d� ∫

∞

0d! (!)Xj(−� ) sin!� , (3.92)

with (!) ≡ (!) coth(�!/2).However, with the aim of doing numerical calculations, it is better to work directly

with Eq. (2.19) expressed in the system eigenbasis, {|��⟩}, ful�lling HS |��⟩ = �� |��⟩.Introducing the identity ∑� |��⟩⟨�� |, noting that

⟨�� |Xj(−� )|��⟩ = e−i(��−�� )�⟨�� |Xj |��⟩ , (3.93)


and using the short-hand notation X (j)�� ≡ ⟨�� |Xj |��⟩, and �� ≡ ⟨�� |�|��⟩, we can write:

−⟨�� |[�]|��⟩ = ∑j, �′, �′

∫∞

0d�

[C(� )ei(��′−��′ )�X (j)

��′X (j)�′� ′�� ′� − C(−� )e−i(��′−�� )�X (j)

�� ′X (j)�′�� ′�′

− C(� )ei(��′−�� )�X (j)��′X (j)

�′�� ′�′ + C(−� )e−i(��′−��′ )�X (j)�′� ′X (j)

� ′��′] . (3.94)

Now, we de�ne

Γ(!) ≡ ∫∞

0d� C(� )ei!� =

{ (!) [1 + nB(!)] , ! > 0 (−!)nB(−!) , ! < 0 , (3.95)

and Γ�� ≡ Γ(�� −��). We have neglected the imaginary part of the integral, i.e., the Lamb-Shift, since it only a�ects the coherent part of the dynamics. The bath autocorrelationfunction satis�es C(−� ) = C(� )∗, so we can express Eq. (3.94), as:

⟨�� |[�]|��⟩ = ∑j, �′, �′ [

(Γ∗�′� + Γ� ′� )X (j)�� ′X (j)

�′�

− ��′ ∑�′′Γ� ′�′′X (j)

��′′X (j)�′′� ′ − ��′ ∑

�′′Γ∗�′� ′′X (j)

�′� ′′X (j)� ′′�]��

′�′ . (3.96)

We have rearranged a bit the indices so that we readily identify the matrix form ofthe Liouvillian superoperator , ⟨�� |[�]|��⟩ = ∑� ′�′ ��,� ′�′�� ′�′ . Together with thecoherent part of the evolution, we have the following set of �rst-oder di�erentialequations for the matrix elements of the density matrix:

�� = −i(�� − ��)�� + ∑� ′, �′

��,�′�′�� ′�′ . (3.97)

3.C. Average over pure initial statesAs an ensemble of pure states, we consider all normalized linear combinations | ⟩ =∑N

n=1 cn|n⟩ of orthonormal basis states |n⟩, n = 1,… , N . For the probability distributionof the coe�cients cn, we request invariance under unitary transformations, which leadsto

P (c1,⋯ , cN ) =(N − 1)!�N � (1 − r2) , (3.98)

3.D. TWO-LEVEL SYSTEM DECAY RATES 57

where r2 = ∑Nn=1 |cn|2. This corresponds to an homogeneous distribution on the surface

of a 2N -dimensional unit sphere, while averages of the kind

cnc∗m =1N �nm , (3.99)

cnc∗mcn′c∗m′ = 1N (N + 1)(�nm�n′m′ + �nm′�n′m) , (3.100)

follow from integrals of polynomials over its (2N − 1)-dimensional surface [103]. Con-sequently, we �nd the ensemble averages

tr(�A) = 1N tr(A) , (3.101)

tr(�A�B) = tr(A) tr(B) + tr(AB)N (N + 1) . (3.102)

To compute averages for pure states belonging to a particular subspace of dimensionND , we have to replace N by ND and the operators A and B by their projections ontothat subspace, PDAPD and PDBPD .

3.D. Two-level system decay ratesFor completeness, we summarize the Bloch-Red�eld result for the decay rates of thetwo-level system coupled to an Ohmic bath [99, 100]. Following the notation used inthe main text, its Hamiltonian is de�ned by

H = Δ2 �x +�2�z +

12XB , (3.103)

with the tunnel matrix element Δ and the detuning �. The bath coupling is speci�ed by(i) X = �z for charge noise and (ii) X = �x for current noise, respectively. To establisha relation to our Hubbard chain, we identify the detuning by the interaction, � ∼ U ,and the tunnel coupling by Δ ∼ J . Note that replacing charge noise by current noisecorresponds to changing � → −Δ and Δ→ �. Therefore, we can restrict the derivationof the decay rate to case (ii).

It is straightforward to transform the Hamiltonian into the eigenbasis of the two-level system, where it reads

H ′ = E2�z + X

′B , (3.104)

with E =√�2 + Δ2, while the system-bath coupling becomes

X ′ = �2E�x +

Δ2E�z . (3.105)


In the interaction picture, it is

X (−� ) = 12E [��x cos(E� ) + ��y sin(E� ) + Δ�z] . (3.106)

Again ignoring the imaginary part of the integral in Eq. (3.38), the noise kernel can bewritten as

Q = �2E

(E)2 �x +

Δ2E

(0)2 �z . (3.107)

The projector to the high-energy state is P1 = (�0 + �z)/2, so that the decay rate can befound as

Γii = tr (P1[Q, [X, P1]]) = (�2E)

2(E) . (3.108)

Accordingly, we �nd for case (i) the rate

Γi = (Δ2E)

2(E) . (3.109)

As it turns out, the low-temperature limit for an Ohmic spectral density

(E) ∝ E coth (�E/2) , (3.110)

coincides with the high-temperature limit at kBT = E/2, since coth(x) ∼ x−1 in the limitx → 0.

4Topological qantum optics

The spectacular progress in recent years in the �eld of topological matter has moti-vated the application of topological ideas to the �eld of quantum optics. The startingimpulse was the observation that topological bands also appear with electromagneticwaves [104]. Soon after that, many experimental realizations followed [44]. Nowadays,topological photonics is a burgeoning �eld with many experimental and theoreticaldevelopments. Among them, one of the current frontiers of the �eld is the explorationof the interplay between topological photons and quantum emitters [105–107].

In this chapter we analyze what happens when quantum emitters interact with atopological wave-guide QED bath, namely a photonic analogue of the SSH model, andshow that it causes a number of unexpected phenomena [6*] such as the emergence ofchiral photon bound states and unconventional scattering. Furthermore, we show howthese properties can be harnessed to simulate exotic many-body Hamiltonians.

4.1. �antum emitter dynamics

The system under consideration is depicted schematically in Fig. 4.1, Ne quantumemitters (QEs) interact with a common bath, which behaves as the photonic analogue ofthe SSH model. This bath model consists of two interspersed photonic lattices A/B withalternating nearest neighbour hoppings J (1 ± �). We assume that the A/B modes havethe same energy !c , that from now on we take as the reference energy of the problem,i.e., !c = 0. Since we have already analyzed this model in section 2.1 we do not give anyfurther details here. For the QEs, we consider they all have a single optical transitiong-e with a detuning !e respect to !c , and they couple to the bath locally. Thus, the QEsand the bath are jointly described by the Hamiltonian H = HS + HB + HI , with

HS = !eNe∑n=1

�nee , (4.1)

59

60 CHAPTER 4. TOPOLOGICAL QUANTUM OPTICS

HB = −JN∑j=1

[(1 + �)c†jAcjB + (1 − �)c†j+1AcjB + H.c.] , (4.2)

HI = gNe∑n=1

(�negcxn�n + H.c.) . (4.3)

Here, cjA (cjB) annihilates a photon at the jth unit cell in the A (B) sublattice; xn and �ndenote the unit cell and sublattice to which the nth QE is coupled. We use the notation�n�� = |�⟩n⟨� |, �, � ∈ {e, g} for the nth QE operator. The interaction is treated within therotating-wave approximation such that only number-conserving terms appear in HI .

Figure 4.1: Schematic picture of the system under consideration: One or many two-levelquantum emitters (in blue) interact with the photonic analogue of the SSH model. Theinteraction with photons (in red) induces non-trivial dynamics between them.

To compute the dynamics of the system we can use two di�erent approaches. In theweak-coupling regime the bath can be e�ectively traced out, such that the evolution ofthe QE reduced density matrix � is described by a Markovian master equation [64] (seeappendix 4.B):

� = i[�, HS] + i∑m, n

J ��mn [�, �meg�nge] +∑m, n

Γ��mn2 [2�nge��meg − �meg�nge� − ��meg�nge] . (4.4)

The functions J ��mn and Γ��mn, which control the QE coherent and dissipative dynamics, arethe real and imaginary parts of the collective self-energy Σ��mn(!e + i0+) = J ��mn − iΓ��mn/2.This collective self-energy depends on the sublattices �, � ∈ {A, B} to which the mthand nth QE couple respectively, as well as on their relative position xmn = xn − xm.Remarkably, for our model they can be calculated analytically in the thermodynamic

4.1. QUANTUM EMITTER DYNAMICS 61

limit (N → ∞) yielding (see appendix 4.A):

ΣAA/BBmn (z) = −g2z [y |xmn |+ Θ(1 − |y+|) − y |xmn |− Θ(|y+| − 1)]√

z4 − 4J 2(1 + �2)z2 + 16J 4�2, (4.5)

ΣABmn(z) =g2J [Fxmn (y+)Θ(1 − |y+|) − Fxmn (y−)Θ(|y+| − 1)]√

z4 − 4J 2(1 + �2)z2 + 16J 4�2, (4.6)

where Fn(z) = (1 + �)z |n| + (1 − �)z |n+1|, Θ(z) is Heaviside’s step function, and

y± =z2 − 2J 2(1 + �2) ±

√z4 − 4J 2(1 + �2)z2 + 16J 4�2

2J 2(1 − �2) . (4.7)

When the transition frequency of the emitters lays in one of the bath’s energybands, generally Γ��mn ≠ 0, so the Markovian approximation predicts the decay of thoseemitters that are excited, emitting a photon into the bath. On the other hand, if thetransition frequency lays in one of the band gaps, Γ��mn = 0, so the emitters will notdecay, but they will interact with each other through the emission and absorption ofvirtual photons in the bath, that is, the bath mediates dipolar interactions between theemitters. However, since we have a highly structured bath, this perturbative descriptionwill not be valid in certain regimes, e.g., close to band-edges, and we will use resolventoperator techniques [66] to solve the problem exactly for in�nite bath sizes (see section2.3.2 for a brief introduction to the resolvent operator formalism).

4.1.1. Single emitter dynamicsLet us begin analyzing the dynamics of a single QE coupled to the bath. If the QE isinitially excited, the wavefunction of the system at time t = 0 is given by | (0)⟩ =|e⟩|vac⟩ (|vac⟩ denotes the vacuum state of the bath). Since the Hamiltonian conservesthe number of excitations, the wavefunction of the system at any later time has theform:

| (t)⟩ = [ e(t)�eg +∑j ∑�=A,B

j� (t)c†j�] |g⟩|vac⟩ . (4.8)

The probability amplitude e(t) can be computed as the Fourier transform of thecorresponding matrix element of the resolvent, i.e., the emitter’s Green’s functionGe = [z − !e − Σe(z)]−1,

e(t) =−12�i ∫

∞

−∞dE Ge(E + i0+)e−iEt , (4.9)

which depends on the emitter self-energy Σe ,

Σe(z) =g2z sign(|y+| − 1)√

z4 − 4J 2(1 + �2)z2 + 16J 4�2, (4.10)


obtained from Eq. (4.5) de�ning Σe(z) ≡ ΣAAnn (z). As we can see, Σe does not depend onthe sign of � . Thus, the single emitter dynamics is insensitive to the bath’s topology.

�3 �2 �1 0 1 2 3

!e ŒJ �

�0:5

0:0

0:5

1:0ı!e;�eŒJ�

Figure 4.2: Comparison between the exact decay rate given by the imaginary part ofthe complex poles of Ge (orange circles) and the Markovian decay rate (purple line)as a function of the emitter’s bare frequency. We also plot the Markovian Lamb shift(blue line). The dashed vertical lines mark the position of the bath’s band edges. Theparameters of the system are � = 0.5, g = 0.4J .

To compute the integral in Eq. (4.9), we can use residue integration closing thecontour of integration in the lower half of the complex plane. As a �rst approximation,one can assume that Σe is small and varies little in the neighbourhood of !e , andsubstitute z by !e in its argument. This is essentially the same as the Markovianapproximation, since the Green’s function has then a single pole at !e + Σe(!e + i0+)and the evolution is given by e(t) = e−i[!e+Σe (!e+i0+)]t . One can readily identify �!e ≡Re Σe(!e + i0+) as a shift of the emitter’s frequency, known as the Lamb shift, andΓe ≡ −2 ImΣe(!e + i0+) as the decay rate of the emitter’s excited state, which canbe expressed in terms of the bath’s density of modes D(E) at emitter frequency asΓe = �g2D(!e) (Fermi’s golden rule). They are plotted in Fig. 4.2. Note that Γe is nonzeroonly inside the bath’s band regions, while the �!e is nonzero only outside them. Thisresult corresponds to the basic expectation that if the frequency of the emitter is withinthe bath’s energy bands of allowed modes, the emitter will decay exponentially witha decay rate given by Fermi’s golden rule. On the contrary, if the emitter’s frequencylays outside the bath’s energy bands, it will remain excited.

As we will see next, an exact calculation of the integral yields somewhat di�erentresults. It requires choosing a proper contour of integration due to the branch cutsthat the Green’s function has along the real axis in the regions where the bands of thebath are de�ned. A way to do it is to take a detour at the band edges to other Riemannsheets of the function, see Fig. 4.3. The formula for the Green’s function in the �rstRiemann sheet G I

e is the one we have provided already. Its analytical continuation to


the second Riemann sheet G IIe can be obtained changing the sign of the square root in

the denominator of Σe .

Bound states

Unstable

pole

Branch cut detour

Figure 4.3: Integration path to compute the dynamics (in blue). Since the Green’sfunction has branch cuts along the real axis, it is necessary to take a detour to thesecond Riemann sheet of the function (shaded areas). The dynamics can be split in thecontribution from the real poles of the function (black dots), the complex poles (whitedot) and the detours taken at the band edges. G I

e has no complex poles in the lower halfcomplex plane thus we only have to consider them in the band regions where we go tothe second Riemann sheet. Real poles only appear in the band gap regions.

According to this procedure the dynamics can be split in contributions of threedi�erent kinds:

e(t) =∑zBSR(zBS)e−izBSt +∑

zUPR(zUP)e−izUPt +∑

j BC,j(t) . (4.11)

The �rst term accounts for the contribution of real poles of Ge , the so-called bound states(BS). These are non-decaying solutions of the Schrödinger equation. The second termaccounts for the contribution of unstable poles (UP), i.e., complex poles of Ge . These aresolutions that decay exponentially. The residue at both the real and complex poles canbe computed as R(z0) = [1 − Σ′e(z0)]−1, where Σ′e(z0) denotes the �rst derivative of theappropriate function ΣIe(z) or ΣIIe (z). It can be interpreted as the overlap between theinitial wavefunction and these solutions. Finally, we should subtract the detours takendue to the branch cuts. Their contribution can be computed as

BC,j(t) =±12� ∫

∞

0dy [G I

e (xj − iy) − G IIe (xj − iy)] e−i(xj−iy)t , (4.12)

with xj ∈ {±2J , ±2|� |J}. The sign has to be chosen positive if when going from xj + 0+to xj − 0+ the integration goes from the �rst to the second Riemann sheet, and negativeif it is the other way around.


We can now point out several di�erences between the exact dynamics and thedynamics within the Markovian approximation. To begin with, the actual decay ratedoes not diverge at the band edges, but acquires a �nite value, contrary to the Markovianprediction, see Fig. 4.2. Furthermore, the actual decay is not purely exponential, asthe BC contributions decay algebraically ∼ t−3 [108] (see appendix 4.C). In Fig. 4.4(a)it is shown an example of this subexponential decay when !e is placed at the lowerband edge of the bath’s spectrum. Another di�erence between the Markovian and theexact result is the fractional decay that the emitter experiences when its frequency laysoutside the bath’s energy bands . This is due to the emergence of photon bound stateswhich localize the photon around the emitter [109–111]. For example, in Fig 4.4(b) weshow the dynamics of an emitter with frequency in the middle of the band gap (!e = 0).It remains in the excited state with a long time limit given by the residue at zBS = 0,limt→∞ | e(t)|2 = |R(0)|2 = [1 + g2/(4J 2|� |)]

−2.

101 102 103

time�J�1

�

10�6

10�4

10�2

100

jD.t/j2 / t�3

(a)

0 25 50 75 100

time�J�1

�0:0

0:2

0:4

0:6

0:8

1:0

j e.t/j2

jıj D 0:70:5

0:3

0:1

0:05

(b)

Figure 4.4: (a) Sub-exponential decay for a system with parameters: !e = −2J , � =0.5 and g = 0.2J . We plot the squared absolute value of the decaying part of thedynamics D(t) = e(t) −∑zBS R(zBS)e−izBSt . (b) Fractional decay for di�erent values of thedimerization parameter. The rest of parameters of the system are !e = 0 and g = 0.4J .As the band gap closes (� → 0) the decay becomes stronger. The dashed lines mark thevalue of |R(0)|2

These photon bound states are not unique to this particular bath [112]. However,the BSs appearing in the present topological waveguide bath have some distinctivefeatures with no analogue in other systems, and deserve special attention. As we willsee later, they play a crucial role in the coherent evolution of many emitters. We can �ndtheir energy and wavefunction solving the secular equation H |ΨBS⟩ = EBS|ΨBS⟩, withEBS outside the band regions and |ΨBS⟩ in the form of Eq. (4.8) with time-independentcoe�cients. Without loss of generality we assume that the emitter couples to sublattice


A at the j = 0 cell. After some algebra, one �nds that the energy of the BS is given bythe pole equation EBS = !e + Σe(EBS). Irrespective of the values of !e or g, there arealways three solutions to this equation. This is because the self-energy diverges in allband edges (see Fig. 4.2), which guarantees �nding a BS in each of the band gaps. Thewavefunction amplitudes can be obtained as

jA =gEBS e2� ∫

�

−�dk eikj

E2BS − !2k, (4.13)

jB =g e2� ∫

�

−�dk !ke

i(kj−�k )

E2BS − !2k, (4.14)

where e is a constant obtained from the normalization condition that is directly relatedwith the long-time population of the excited state in spontaneous emission. They areplotted in the left column of Fig. 4.5. From Eqs. (4.13) and (4.14) we can extract severalproperties of the spatial wavefunction distribution. On the one hand, above or below thebands (|EBS| > 2J , upper and lower band gaps) the largest contribution to the integralsis that of k = 0. Thus, the amplitude of the wavefunction in any sublattice j� has thesame sign regardless the unit cell j. In the lower (upper) band-gap, the amplitude onthe di�erent sublattices has the same (opposite) sign. On the other hand, in the innerband gap (|EBS| < 2|� |J ), the main contribution to the integrals is that of k = � . Thisgives an extra factor (−1)j to the coe�cients j� . Furthermore, in any band gap, theamplitudes on the sublattice to which the QE couples are symmetric with respect tothe position of the QE, whereas they are asymmetric in the other sublattice, that is, theBSs are chiral. Changing � from positive to negative results in a spatial inversion ofthe BS wavefunction. The asymmetry of the BS wavefunction is more extreme in themiddle of the inner band-gap, for !e = 0. At this point the BS has energy EBS = 0. If� > 0, its wavefunction is given by jA = 0 and

jB =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

g e(−1)jJ (1 + �) (

1 − �1 + �)

j, j ≥ 0

0 , j < 0, (4.15)

whereas for � < 0 the wavefunction decays for j < 0, and is strictly zero for j ≥ 0. Itsdecay length diverges as �BS ∼ 1/(2|� |) when the gap closes. Away from this point, theBS decay length shows the usual behavior for 1D baths �BS ∼ |Δedge|−1/2, with Δedge beingthe smallest detuning between the QE frequency and the band-edges.

The physical intuition behind the appearance of such chiral BS at EBS = 0 is thatthe QE with !e = 0 acts as an e�ective edge in the middle of the chain, or equivalently,as a boundary between two semi-in�nite chains with di�erent topology. In fact, onecan show that this chiral BS has the same properties as the edge-state that appears in asemi-in�nite SSH chain in the topologically non-trivial phase, for example, inheriting


�0:2

0:0

0:2

no disorder o�-diagonal diagonal

�0:2

0:0

0:2

jA; jB

�5 0 5

�0:3

�0:2

�0:1

0:0

�5 0 5

j

�5 0 5

EBS ' 2:33J EBS ' 2:36J EBS ' 2:34J

EBS ' 0:0J EBS ' 0:0J EBS ' �0:02J

EBS ' �2:33J EBS ' �2:32J EBS ' �2:33J

Figure 4.5: Single-photon bound states for a single emitter with transition frequencies!e = 2.2J (upper row), !e = 0 (middle row) and !e = −2.2J (bottom row). Only thedominant bound state is plotted for each case. Rest of parameters: g = 0.4J , � = 0.2,and w = 0.6J for the disordered cases

its robustness to disorder. To illustrate it, we study the e�ect of two types of disorder:one that appears in the cavities’ bare frequencies, and another one that appears in thetunneling amplitudes. The former corresponds to the addition of random diagonalterms to the bath’s Hamiltonian and breaks the chiral symmetry of the original model,

HB → HB +∑j

∑�=A,B

�j�c†j�cj� , (4.16)

while the latter corresponds to the addition of o�-diagonal random terms and preserves


it,

HB → HB +∑j(�j1c†jBcjA + �j2c†j+1AcjB + H.c.) . (4.17)

We take the coe�cients �j� , � ∈ {A, B, 1, 2}, from a uniform distribution within therange [−w/2, w/2]. To prevent changing the sign of the coupling amplitudes betweenthe cavities, w is restricted to w/2 < J (1 − |� |) in the case of o�-diagonal disorder.

In the middle (right) column of Fig. 4.5 we plot the shape of the three BS appearing inour problem for a situation with o�-diagonal (diagonal) disorder with w = 0.6J . There,we observe that while the upper and lower BS get modi�ed for both types of disorder,the chiral BS has the same protection against o�-diagonal disorder as a regular SSHedge-state: its energy is pinned at EBS = 0 as well as keeping its shape with no amplitudein the sublattice to which the QE couples to. On the contrary, for diagonal disorder themiddle BS is not protected any more and may have weight in both sublattices.

Finally, to make more explicit the di�erent behavior with disorder of the middleBS compared to the other ones, we compute their localization length �BS as a functionof the disorder strength w averaging for many realizations. In Fig. 4.6 we plot boththe average value (markers) of �−1BS and its standard deviation (bars) for the cases ofthe middle (blue circles) and upper (purple triangles) BSs. Generally, one expects thatfor weak disorder, states outside the band regions tend to delocalize, while for strongdisorder all eigenstates become localized (see, for example, Ref. [1*]). In fact, this isthe behavior we observe for the upper BS for both types of disorder. However, thenumerical results suggest that for o�-diagonal disorder the chiral BS never delocalizes(on average). Furthermore, the chiral BS localization length is less sensitive to thedisorder strength w manifested in both the large initial plateau as well as the smallerstandard deviations compared to the upper BS results.


0:0 0:5 1:0

w ŒJ �

0:5

1:0

1:5

1=�

BS

o�-diagonal

0 2 4

w ŒJ �

diagonal

Figure 4.6: Inverse BS localization length as a function of the disorder strength w forboth diagonal and o�-diagonal disorder. The markers correspond to the average valuecomputed with a total of 104 instances of disorder, and the error bars mark the value ofone standard deviation above and below the average value. The two sets of points areslightly o�set along the x axis for better visibility. The two cases shown correspondto !e ≃ 2.06J (purple triangles) and !e = 0 (blue circles), which in the case withoutdisorder have the same decay length. The rest of parameters are: g = 0.4J and � = 0.5.

4.1.2. Two emitter dynamicsThe next simplest case we can study is that of two QE coupled to the bath. From themaster equation (4.4), we can see that if the QEs frequency is in one of the band gaps,the interaction with the bath leads to an e�ective unitary dynamics governed by thefollowing Hamiltonian:

Hdd = J ��12 (� 1eg� 2ge + H.c.) . (4.18)

That is, the bath mediates dipole-dipole interactions between the emitters. One wayto understand the origin of these interactions is that the emitters exchange virtualphotons through the bath. In fact, these virtual photons are nothing but the photonBS that we have studied in the previous section. Thus, these interactions J ��mn inheritmany properties of the BSs. For example, the interactions are exponentially localizedin space, with a localization length that can be tuned and made large by setting !eclose to the band-edges, or �xing !e = 0 and letting the middle band gap close, � → 0.Moreover, one can also change qualitatively the interactions by moving !e to di�erentband gaps: for |!e | > 2J all the J ��mn have the same sign, while for |!e | < 2|� |J theyalternate sign as xmn increases. Also, changing !e from positive to negative changesthe sign of JAA/BBmn , but leaves unaltered JAB/BAmn . Furthermore, while JAA/BBmn are insensitiveto the bath’s topology, the JAB/BAmn mimic the dimerization of the underlying bath, butallowing for longer range couplings. The most striking regime is reached for !e = 0.In that case JAA/BBmn identically vanish, so the QEs only interact if they are coupled to


di�erent sublattices. Furthermore, in such a situation the interactions have a strongdirectional character, i.e., the QEs only interact if they are in some particular order.Assuming that the �rst QE at x1 couples to sublattice A, and the second one at x2 couplesto B, we have

JAB12 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

sign(�) g2(−1)x12J (1+�) ( 1−�1+� )x12 , � ⋅ x12 > 0

0 , � ⋅ x12 < 0Θ(�) g2

J (1+�) , x12 = 0. (4.19)

We can also apply resolvent operator techniques to compute the dynamics of twoemitters exactly. It can be shown that the symmetric and antisymmetric combina-tions �†± = (� 1eg ± � 2eg) /

√2 evolve independently as they couple to orthogonal bath

modes [113] (see appendix 4.A). Thus, the two-emitter problem is equivalent to twoindependent single-emitter problems. Remarkably the self-energies adopt the simpleform Σ��± = Σe ± Σ��mn. We can now compute the two-emitter BSs’ energies solvingthe pole equations EBS,± = !e + Σ��± (EBS,±). However, there are some subtleties whichdi�erentiate this problem from the single-emitter problem. Now, the cancellation ofdivergences in Σ± at the band edges results in critical values for the emitter transitionfrequency above (or below) which some bound states cease to exist. For example, fortwo emitters in the AB con�guration, in the symmetric subspace we have that the lowerbound state (EBS,+ < −2J ) always exists, while the upper bound state (EBS,+ > 2J ) existsonly for !e > !crit,

!crit = 2J −g2(2x12 + 1 − �)2J (1 − �2) . (4.20)

For the middle bound state, there are two possibilities: either the divergence vanishes at−2|� |J , in which case the bound state will exist for !e > !crit, or the divergence vanishesat 2|� |J , then the middle bound state exists for !e < !crit. In both cases !crit takes thesame form

!crit = (−1)x12{2�J + g

2[(2x12 + 1)� − 1]2J (1 − �2)

}. (4.21)

The situation in the antisymmetric subspace can be readily understood realizing thatRe Σ��− (z) = − Re Σ��+ (−z), which implies that if EBS,+ is a solution of the pole equation forΣ��+ for a particular value of !e , then EBS,− = −EBS,+ is a solution of the pole equation forΣ��− for the opposite value of !e . Fig. 4.7 summarizes at a glance the di�erent possibilitiesand the dependence on the bath’s topology.

The physical intuition behind this phenomenon is the following: when bringingclose together two emitters, their respective single-emitter bound states hybridizeforming (anti)symmetric superpositions which have energies above and below the


single emitter bound state energy. Back in the localized basis the splitting of the boundstate energies corresponds to the “hopping” of the photon, i.e., an e�ective dipole-dipoleinteraction between the emitters, J ��12 = (EBS,+ − EBS,−)/2. If this splitting is very strong,it may happen that one of the two bound states merges into the bulk bands. Then itis not possible to rewrite the low-energy Hamiltonian as an e�ective dipole-dipoleinteraction [114]. In Fig. 4.8 we show the exact value of the interaction constant andcompare it with the Markovian result (4.19). Apart from small deviations when thegap closes for � → 0, it is important to highlight that the directional character agreesperfectly in both cases.

�4 �2 0 2 4

!e ŒJ �

�4

�2

0

2

4

EBS;C;E

BS;�ŒJ�

topological

�4 �2 0 2 4

!e ŒJ �

trivial

Figure 4.7: Exact energies for the symmetric (continuous line) and antisymmetric(dashed line) bound states for a system with parameters: g = 0.8J , and � = 0.5 (right) or� = −0.5 (left). The two emitters are in the AB con�guration coupled to the same unitcell, x12 = 0. The grey areas mark the span of the bath’s energy bands.

When the QEs frequency is resonant with one of the bath’s bands, the bath typicallyinduces non-unitary dynamics in the emitters. However, when many QEs couple to thebath there are situations in which the interference between their emission may enhanceor suppress (even completely) the decay of certain states. This phenomenon is knownas super/subradiance [115], respectively. Let us illustrate this e�ect with two QEs: Inthat case, the decay rate of a symmetric/antisymmetric combination of excitations isΓe ± Γ��12 . When Γ��12 = ±Γe , these states decay at a rate that is either twice the individualone or zero. In this latter case they are called perfect subradiant or dark states.

In standard one-dimensional baths Γ12(!e) = Γe(!e) cos (k(!e)|xmn|), so the darkstates are such that the wavelength of the photons involved, k(!e), allows for theformation of a standing wave between the QEs when both try to decay, i.e., whenk(!e)|xmn| = n� , with n ∈ ℕ. Thus, the emergence of perfect super/subradiant statessolely depends on the QE frequency !e , bath energy dispersion !k , and their relative


Figure 4.8: E�ective dipolar coupling as given by the BSs energy di�erence (dots), andthe Markovian approximation (4.19) (lines) as a function of the dimerization constant.The rest of parameters are !e = 0 and g = 0.4J . The schematics above show the shapeof the bound states in the topological (right) and trivial (left) phases. The situation forthe BA con�guration is the same, reversing the role of � .

position xmn, which is the common intuition for this phenomenon. This commonwisdom gets modi�ed in the topological bath that we have considered, where we �ndsituations in which, for the same values of xmn, !k and !e , the induced dynamics is verydi�erent depending on the sign of � . In particular, when two QEs couple to the A andB sublattice respectively, the collective decay reads:

ΓAB12 (!e) = Γe sign(!e) cos (k(!e)x12 − �(!e)) , (4.22)

which depends both on the photon wavelength mediating the interaction

k(!e) = arccos(!2e − 2J 2(1 + �2)2J 2(1 − �2) ) , (4.23)

an even function of � , and on the phase �(!e) ≡ �(k(!e)), which is sensitive to thesign of � . This �-dependence enters through the system-bath coupling when rewritingthe interaction Hamiltonian HI [Eq. (4.3)] in terms of the eigenoperators uk , lk (seeappendix 4.A). Thus, even though the sign of � does not play a role in the propertiesof an in�nite bath, when the QEs couple to it, the bath embedded between them isdi�erent for � ≷ 0, making the two situations inequivalent.

Using Eq. (4.22), we �nd that the detunings at which perfect super/subradiant statesappear satisfy k(!s)x12 − �(!s) = n� , n ∈ℕ. They come in pairs: If !s corresponds to


a superradiant (subradiant) state in the upper band, −!s corresponds to a subradiant(superradiant) state in the lower band. In particular, it can be shown that when � < 0,the previous equation has solutions for n = 0,… , x12, while if � > 0, the equation hassolutions for n = 0,… , x12 + 1. Besides, the detunings, !s at which the subradiant statesappear also satisfy that JAB12 (!s) ≡ 0, which guarantees that these subradiant statessurvive even in the non-Markovian regime with a correction due to retardation whichis small as long as x12Γe(!e)/(2|vg(!e)|) ≪ 1 [vg(!e) is the group velocity of the photonsin the bath at frequency !e] [113]. Apart from inducing di�erent decay dynamics,these di�erent conditions for super/subradiance at �xed !e also translate in di�erentre�ection/transmission coe�cients when probing the system through photon scattering,as we show in the next section.

4.2. Single-photon scatteringHere, we study the case when the emitter frequencies are resonant with one of thebands. We will see how for a single emitter coupled to both the A and B cavities insidea unit cell there is a �-dependent Lamb-Shift that can be detected in single-photonscattering experiments. Also, we will see how the di�erent super/subradiant states for±� lead to di�erent behavior when a single photon scatters o� two QEs.

4.2.1. Scattering formalismThe scattering properties of a single photon impinging into one or several QEs in theground state can be obtained from the scattering eigenstates, which are solutions ofthe secular equation H |Ψk⟩ = ±!k |Ψk⟩ (the sign depends on the band we are probing).First, let us assume that a single emitter couples to the A sublattice at the x1 unit cell.We use the ansatz

|Ψk⟩ ={ ink |k⟩ + out

−k |−k⟩ , j < x1 outk |k⟩ + in

−k |−k⟩ , j ≥ x1, (4.24)

where |±k⟩ = u†±k |vac⟩ or |±k⟩ = l†±k |vac⟩, depending on the band we are probing (uk andlk are the eigenmodes of the SSH model, see section 2.1). For a schematic representationof |Ψk⟩ see Fig. 4.9.

The coe�cients of the scattering eigenstate in position representation are then

jA = ±{ ink ei(kj+�k ) + out

−k e−i(kj+�k ) , j ≤ x1 outk ei(kj+�k ) + in

−ke−i(kj+�k ) , j ≥ x1, (4.25)

jB ={ ink eikj + out

−k e−ikj , j < x1 outk eikj + in

−ke−ikj , j ≥ x1. (4.26)

4.2. SINGLE-PHOTON SCATTERING 73

Figure 4.9: Schematic picture of a scattering eigenstate and the di�erent amplitudesinvolved. The QE divides the space in left and right regions. Incoming modes arethose that propagate towards the emitter (red), while outgoing modes are those thatpropagate away from the emitter (black).

The matching condition at the emitter position

ink ei(kx1+�k ) + out

−k ei(kx1+�k ) = outk ei(kx1+�k ) + in

−kei(kx1+�k ) , (4.27)

together with the secular equation{!e e + g x1A = ±!k e ,g e + −J (1 + �) x1B − J (1 − �) x1−1B = ±!k x1A ,

(4.28)

⇒ −J (1 + �) x1B − J (1 − �) x1−1B = (±!k −g2Δk)

x1A , (4.29)

where we have de�ned Δk ≡ ±!k − !e , allow us to write a linear relation between thewave amplitudes on the left of the emitter and those on the right as

( outk in−k )

= T ( ink

out−k ) , (4.30)

where the transfer matrix T is

TA =

⎛⎜⎜⎜⎜⎝

1 ± g2i2ΔkJ (1 − �) sin(k + �k)

±g2e−i2(kx1+�k )i2ΔkJ (1 − �) sin(k + �k)

∓g2ei2(kx1+�k )i2ΔkJ (1 − �) sin(k + �k)

1 ∓ g2i2ΔkJ (1 − �) sin(k + �k)

⎞⎟⎟⎟⎟⎠

. (4.31)

Similarly, if the emitter couples to the B sublattice at the x1 unit cell we have

TB =

⎛⎜⎜⎜⎜⎝

1 ∓ g2i2ΔkJ (1 + �) sin(�k)

∓g2e−i2kx1i2ΔkJ (1 + �) sin(�k)

±g2ei2kx1i2ΔkJ (1 + �) sin(�k)

1 ± g2i2ΔkJ (1 + �) sin(�k)

⎞⎟⎟⎟⎟⎠

. (4.32)


From the transfer matrix we can compute the scattering matrix S, which relates theasymptotic incoming modes with the outgoing modes:

( outk out−k ) = S (

ink

in−k)

, (4.33)

with

S =⎛⎜⎜⎜⎝

t11 −t12t21t22

t12t22

− t21t221t22

⎞⎟⎟⎟⎠

≡ (tL rRrL tR) . (4.34)

Here, tij denote the matrix elements of the transfer matrix, while tL/R and rL/R denotethe matrix elements of the scattering matrix. They correspond to the transmission andre�ection probability amplitudes for a wave coming from the left/right.

If evolution is unitary, that is, there are no photon losses, S†S = SS† = I , whichimplies |tL|2 + |rL|2 = |tR |2 + |rR |2 = 1 and |tL|2 + |rR |2 = |tR |2 + |rL|2 = 1. Therefore, |tL| = |tR |and |rL| = |rR |. Furthermore if the system is time-reversal symmetric (H is real), as isthe case in our model, the scattering is reciprocal, i.e., tL = tR ≡ t . To see this, let usconsider the scattering eigenstate with amplitudes ( in

k , in−k , out

k , out−k ) = (1, 0, tL, rL),

and call it |Ψk,L⟩. Then, its complex conjugate is also a scattering eigenstate with thesame energy and so is the linear combination (1/t ∗L)|Ψk,L⟩∗ − (r ∗L/t ∗L)|Ψk,L⟩, which hascoe�cients (0, 1, −r ∗LtL/t ∗L, tL), but this must be the scattering eigenstate with coe�cients(0, 1, rR , tR).

The scattering coe�cients for the many-emitter case can be readily obtained notingthat if we label the emitters with an increasing index from left to right, the �elds on theright of the mth emitter are those on the left of the (m + 1)th emitter. Thus, the transfermatrix of the entire system can be written as the product of single-emitter transfermatrices T = TNeTNe−1… T1 (Ne is the number of emitters) and from it one can computethe scattering matrix of the entire system.

4.2.2. Scattering off one and two emittersFor a single emitter, we �nd the same transmission coe�cient regardless the sublatticeto which the emitter is coupled

t = 2J 2(1 − �2)Δk sin(k)2J 2(1 − �2)Δk sin(k) ∓ ig2!k

. (4.35)

A well-known feature for this type of system is the perfect re�ection (|r |2 = 1⇔ |t |2 = 0)when the frequency of the incident photon matches exactly that of the QE [116]. Thiscan be seen in Fig. 4.10(a) as a full dip in the transmission probability at Δk = 0. The

4.2. SINGLE-PHOTON SCATTERING 75

dip has a bandwidth determined by the individual decay rate Γe . Besides, it also showsthe vanishing of the transmission at the band edges. Since there is no dependence onthe sign of � , the scattering in this con�guration is insensitive to the bath’s topology.

A more interesting situation occurs when a single emitter couples to both the Aand B cavities in a single cell. We choose the coupling constants g� and g(1 − �), suchthat we can interpolate between the cases where the QE couples only to sublattice A(� = 1) or B (� = 0). Using the same ansatz as in the previous case, we �nd

t = 2iJ (1 − �) sin(k) [J (1 + �)Δk − g2�(1 − �)]2iJ 2(1 − �2)Δk sin(k) + g2!k [2�(1 − �)(e−i�k ∓ 1) ± 1]

. (4.36)

Now, for 0 < � < 1, the transmission is di�erent for ±� . In Fig. 4.10(a) we plot thisformula for � = ±0.3, and show that the transmission dip gets shifted. This is due to a�-dependent Lamb-Shift �!e = g2�(1 − �)/ [J (1 + �)]. Notice that Eq. (4.36) is invariantunder the transformation � → 1 − � .

For two emitters coupled equally to the bath at unit cells x1 and x2, in the ABcon�guration, we �nd

t = [2J 2(1 − �2)Δk sin(k)]2

g4!2kei2(kx12−�k ) − [g2!k ± 2iJ 2(1 − �2)Δk sin(k)]2

, (4.37)

whose squared absolute value is plotted in Fig. 4.10(b) for � = ±0.5. The di�erencebetween bath in the topological and trivial phases is more pronounced than in thesingle-emitter case, since now the transmission is qualitatively di�erent in each case:While the case with � > 0 features a single transmission dip at the QEs frequency, for� < 0, the transmission dip is followed by a window of frequencies with perfect photontransmission, i.e., |t |2 = 1. We can understand this behavior realizing that a singlephoton only probes the (anti)symmetric states in the single excitation subspace |S⟩/|A⟩,with the following energies renormalized by the bath, !S/A = !e ± JAB12 , and linewidthsΓS/A = Γe ± Γ12. For the parameters chosen, it can be shown that for � > 0 the QEsare in a perfect super/subradiant con�guration in which one of the states decoupleswhile the other has a 2Γe decay rate. Thus, at this con�guration, the two QEs behavelike a single two-level system with an increased linewidth. On the other hand, when� < 0, both the (anti)symmetric states are coupled to the bath, such that the systemis analogous to a V-type system where perfect transmission occurs for an incidentfrequency ±!EIT = (!SΓA − !AΓS) /(ΓA − ΓS) [117]


Figure 4.10: Relevant level structure and transmission probability for a single photonscattering o� (a) one QE coupled to both A and B cavities inside a unit cell, (b) twoQEs in an AB con�guration separated a distance of x12 = 2 unit cells; |gg⟩ = |g⟩1|g⟩2denotes the common ground state, while |S/A⟩ = (|e⟩1|g⟩2 ± |g⟩1|e⟩2) /

√2 denotes the

(anti)symmetric excited state combination of the two QEs. The parameters in (a) areg = 0.4J , � = ±0.5, !e = 1.5J , and � = 0 or � = 0.3. The dashed orange line correspondsto the case where the emitter couples to a single sublattice (� = 0, 1), it does not dependon the sign of � . The parameters in (b) are g = 0.1J , � = ±0.5, and !e ≃ 1.65J , for whichthe two QEs are in a subradiant con�guration if � > 0. The black dashed line marks thevalue of !EIT.

4.3. Many emitters: effective spin models

One of the main interests of having a platform with BS-mediated interactions is toinvestigate spin models with long-range interactions [118, 119]. The study of thesemodels has become an attractive avenue in quantum simulation because long-rangeinteractions are the source of non-trivial many-body phases [120] and dynamics [121],and are also very hard to treat classically.

Let us now investigate how the shape of the QE interactions inherited from thetopological bath translate into di�erent many-body phases at zero temperature ascompared to those produced by long-range interactions appearing in other setups

4.3. MANY EMITTERS: EFFECTIVE SPIN MODELS 77

such as trapped ions [120, 121], or standard waveguide setups. For that, we considerhaving Ne emitters equally spaced and alternatively coupled to the A/B lattice sites.After eliminating the bath, and adding a collective �eld with amplitude � to control thenumber of spin excitations, the dynamics of the emitters (spins) is e�ectively given by:

Hspin = ∑m, n

JABmn (�m,Aeg �n,Bge + H.c.) −�2∑n (�n,Az + �n,Bz ) , (4.38)

denoting by �n,�� , � = x, y, z, the corresponding Pauli matrix acting on the � ∈ {A, B}site in the nth unit cell. The J ��mn are the spin-spin interactions derived in the previoussubsection, whose localization length, denoted by � , and functional form can be tunedthrough system parameters such as !e .

For example, when the lower (upper) BS mediates the interaction, the J ��mn has nega-tive (alternating) sign for all sites, similar to the ones appearing in standard waveguidesetups. When the range of the interactions is short (nearest neighbor), the physics is welldescribed by the ferromagnetic XY model with a transverse �eld [122], which goes from afully polarized phase when |�| dominates to a super�uid one in which spins start �ippingas |�| decreases. In the case where the interactions are long-ranged the physics is similarto that explained in Ref. [120] for power-law interactions (∝ 1/r3). The longer rangeof the interactions tends to break the symmetry between the ferro/antiferromagneticsituations and leads to frustrated many-body phases. Since similar interactions alsoappear in other scenarios (standard waveguides or trapped ions), we now focus on themore di�erent situation where the middle BS at !e = 0 mediates the interactions, suchthat the coe�cients JABmn have the form of Eq. (4.19).

In that case, the Hamiltonian Hspin of Eq. (4.38) is very unusual: i) spins only interactif they are in di�erent sublattices, i.e., the system is bipartite ii) the interaction is chiralin the sense that they interact only in case they are properly sorted, i.e., the one inlattice A to the left/right of that in lattice B, depending on the sign of � . Note that � alsocontrols the interaction length � . In particular, for |� | = 1 the interaction only occursbetween nearest neighbors, whereas for � → 0, the interactions become of in�niterange. These interactions translate into a rich phase diagram as a function of � and �,which we plot in Fig. 4.11 for a small chain with Ne = 20 emitters (obtained with exactdiagonalization). Let us guide the reader into the di�erent parts:

(1) The region with maximum average magnetization (in white) corresponds to theregimes where � dominates such that all spins are aligned upwards.

(2) Now, if we decrease � from this fully polarized phase in a region where thelocalization length is short, i.e., � ≈ 0.1, we observe a transition into a state withzero average magnetization. This behavior can be understood because in thatshort-range limit JABmn only couples nearest neighbor AB sites, but not BA sites asshown in the scheme of the lower part of the diagram for � > 0 (the opposite is


true for � < 0). Thus, the ground state is a product of nearest neighbor singlets(for J > 0) or triplets (for J < 0). This state is usually referred to as Valence-BondSolid in the condensed matter literature [123]. Note, the di�erence between � ≷ 0is the presence (or not) of uncoupled spins at the edges.

(3) However, when the bath allows for longer range interactions (� > 1), the transitionfrom the fully polarized phase to the phase of zero magnetization does not occurabruptly but passing through all possible intermediate values of the magnetization.Besides, we also plot in Fig. 4.12 the spin-spin correlations along the x and zdirections (note the symmetry in the xy plane) for the case of � = 0 to evidencethat a qualitatively di�erent order appears as � increases. In particular, we showthat the spins align along the x direction with a double periodicity, which wecan pictorially represent by |↑↑↓↓↑↑…⟩x , and that we call double Néel order states.Such orders have been predicted as a consequence of frustration in classicaland quantum spin chains with competing nearest and next-nearest neighbourinteractions [124–126], introduced to describe complex solid state systems suchas multiferroic materials [127]. In our case, this order emerges in a system whichhas long-range interactions but no frustration as the system is always bipartiteregardless the interaction length.

To gain analytical intuition of this regime, we take the limit � → ∞, where theHamiltonian (4.38) reduces to

H ′spin = UHspinU † ≃ J (S+AS−B + H.c.) , (4.39)

where S+A/B = ∑n �n,A/Beg , and the unitary transformation U = ∏n∈ℤodd �n,Az �n,Bz is usedto cancel the alternating signs of JABmn . Equality in Eq. (4.39) occurs for a system withperiodic boundary conditions, while for �nite systems with open boundary conditionssome corrections have to be taken into account due to the fact that not all spins in onesublattice couple to all spins in the other but only to those to their right/left depending onthe sign of � . The ground state is symmetric under (independent) permutations in A andB. In the thermodynamic limit we can apply mean �eld theory, which predicts symmetrybreaking in the spin xy plane. For instance, if J < 0 and the symmetry is broken alongthe spin direction x , the spins will align so that ⟨(SxA)2⟩ = ⟨(SxB )2⟩ = ⟨SxASxB⟩ = (Ne/2)2,and ⟨SxA⟩2 = ⟨SxB⟩2 = (Ne/2)2 .

Since Ne is �nite in our case, the symmetry is not broken, but it is still re�ected inthe correlations, so that

⟨�m,A� �n,A� ⟩ ≃ ⟨�m,A� �n,B� ⟩ ≃ 1/2 , � = x, y . (4.40)

In the original picture with respect to U , we obtain the double Néel order observed inFig. 4.12. As can be understood, the alternating nature of the interactions is crucial for

4.3. MANY EMITTERS: EFFECTIVE SPIN MODELS 79

obtaining this type of ordering. Finally, let us mention that the topology of the bathtranslates into the topology of the spin chain in a straightforward manner: regardlessthe range of the e�ective interactions, the ending spins of the chain will be uncoupledto the rest of spins if the bath is topologically non-trivial.

This discussion shows the potential of the present setup to act as a quantum simu-lator of exotic many-body phases not possible to simulate with other known setups.

VBS

DN

Figure 4.11: Ground state average polarization obtained by exact diagonalization fora chain with Ne = 20 emitters with frequency tuned to !e = 0 as a function of thechemical potential � and the decay length of the interactions � . The di�erent phasesdiscussed in the text, a Valence-Bond Solid (VBS) and a Double Néel ordered phase(DN) are shown schematically below, on the left and right respectively. Interactions ofdi�erent sign are marked with links of di�erent color.


�1

0

1

Cz.r/

� D 0:1 1:5 5:0

0 2 4 6 8 10

r

�1

0

1

Cx.r/

0 2 4 6 8 10

r

�

Figure 4.12: Correlations C� (r) = ⟨� 9� � 9+r� ⟩ − ⟨� 9� ⟩⟨� 9+r� ⟩, � = x, y, z, [Cx (r) = Cy(r)]for the same system as in Fig. 4.11 for di�erent interaction lengths, �xing � = 0 (leftcolumn). Correlations for di�erent chemical potentials �xing � = 5, darker colorscorrespond to lower chemical potentials (right column). Note we have de�ned a singleindex r that combines the unit cell position and the sublattice index. The yellow dashedline marks the value of 1/2 expected when the interactions are of in�nite range.

4.4. SummaryWe have analyzed the dynamics of a set of quantum emitters (two-level systems)whose ground state-excited state transition couples to the modes of a photonic lattice,which acts as a collective structured bath. For this, we have computed analyticallythe collective self-energies, which allowed us to use master equations and resolventoperator techniques to study the dynamics of quantum emitters in the Markovian andnon-Markovian regimes respectively. The behavior depends fundamentally on whetherthe transition frequency of the emitters lays in a range of allowed bath modes or aband gap. If the transition frequency is tuned to a band gap we observe the followingphenomena:

(1) Emergence of chiral bound states, that is, bound states that are mostly localizedon the left or right of the emitter depending on the sign of the dimerizationconstant � of the photonic bath. Speci�cally, when the transition frequency of anemitter lays in the middle of the inner band gap, it acts as a boundary betweentwo photonic lattices with di�erent topology. The resulting bound state has thesame properties as a regular topological edge state of the SSH model including

4.4. SUMMARY 81

the protection against certain types of disorder.

(2) An exact calculation reveals that the existence conditions of the bound states fortwo emitters are di�erent depending on the sign of � .

(3) These bound states give rise to dipolar interactions between the emitters whichdepend on the topology of the underlying bath if the emitters are coupled todi�erent sublattices. In particular, when the transition frequency of the emitterslays in the middle of the inner band gap, the interactions can be toggled on ando� by changing the sign of � .

When the emitters’ frequency is tuned to the band, the topology of the bath re�ectsitself in:

(1) Di�erent super/subradiant conditions depending on the sign of � .

(2) A �-dependent Lamb Shift for a single emitter coupled simultaneously to bothA and B sublattices. This can be detected as a shift of the transmission dip insingle-photon scattering experiments.

(3) Di�erent scattering properties for two emitters in an AB con�guration dependingon the sign of � . This is a consequence of the di�erent super/subradiant conditionsin each phase.

Last, we analyze the zero temperature phases of the e�ective spin Hamiltonians thatcan be generated in the many-emitter case after tracing out the bath degrees of freedom.We �nd that for short-range interactions the emitters realize a valence-bond solid. Onthe other hand, for long-range interactions, the system becomes gapless and a doubleNéel order emerges.

Appendices

4.A. Calculation of the self-energiesTo obtain analytical expressions for the self-energies, it is convenient to �rst expressHI in the bath eigenbasis. For this we just have to invert (2.5) to obtain expresions forthe local operators cj� in terms of uk and lk . Substituting in (4.3) we obtain

HI =g√2N ∑

n∈SA∑kei(kxn+�k )(uk + lk)�neg

+ g√2N ∑

n∈SB∑keikxn (uk − lk)�neg + H.c. , (4.41)

where SA (SB) denotes the set of emitters coupled to the A (B) sublattice.Identifying H0 = HS and V = HB + HI , using P = |e⟩|vac⟩⟨vac|⟨e|, from Eq. (2.33) we

obtain

Σe(z) ≡ ⟨vac|⟨e|R(z)|e⟩|vac⟩ = g22N ∑

k (1

z − !k+ 1z + !k)

. (4.42)

In the thermodynamic limit (N → ∞), the sum can be replaced by an integral, whichcan be computed easily with the change of variable eik = y yielding the result shown inEq. (4.10); the functions y±(z) appearing in the expression are the roots of the polynomial

p(y) = y2 + [2J 2(1 + �2) − z2J 2(1 − �2) ] y + 1 . (4.43)

For the two emitter case, we will �rst show that the (anti)symmetric combinations�†± = (� 1eg ± � 2eg) /

√2 couple to orthogonal bath modes [113]. Substituting �neq, n = 1, 2,

in terms of �†± in Eq. (4.41), and pairing the terms with opposite momentum, we obtainfor the case where the two QEs couple to sublattice A

HAAI = g√

N ∑k>0

∑�=±

√1 + � cos(kx12)(uk� + lk�)�†� + H.c. , (4.44)

uk± = [ei(kx1+�k ) ± ei(kx2+�k )] uk + [e−i(kx1+�k ) ± e−i(kx2+�k )] u−k2√1 ± cos(kx12)

, (4.45)

lk± = [ei(kx1+�k ) ± ei(kx2+�k )] lk + [e−i(kx1+�k ) ± e−i(kx2+�k )] l−k2√1 ± cos(kx12)

. (4.46)


Here, x12 = x2 − x1 is the signed distance between the two emitters. For the case wherethe two QEs are on a di�erent sublattice

HABI = g√

N ∑k>0

∑�=± [

√1 + � cos(kx12 − �k) uk��†�

+√1 − � cos(kx12 − �k) lk��†� ] + H.c. , (4.47)

uk± = [ei(kx1+�k ) ± eikx2] uk + [e−i(kx1+�k ) ± e−ikx2] u−k2√1 ± cos(kx12 − �k)

, (4.48)

lk± = [ei(kx1+�k ) ∓ eikx2] lk + [e−i(kx1+�k ) ∓ e−ikx2] l−k2√1 ∓ cos(kx12 − �k)

. (4.49)

The denominators in the de�nition of uk± and lk± come from normalization. Importantly,these modes are orthogonal, they satisfy

[uk� , u†k′� ′] = [lk� , l†k′�′] = �kk′��′ . (4.50)

Since!k = !−k , we have that the bath Hamiltonian is also diagonal in this new basis. Thetwo other con�gurations, can be analyzed analogously. From these expressions for theinteraction Hamiltonian, it is possible to obtain the self-energy for the (anti)symmetricstates of the two QE,

ΣAA/BB± = g2N ∑

k>0 [1 ± cos(kx12)

z − !k+ 1 ± cos(kx12)z + !k ] , (4.51)

ΣAB± = g2N ∑

k>0 [1 ± cos(kx12 − �k)

z − !k+ 1 ∓ cos(kx12 − �k)z + !k ] . (4.52)

As it turns out, they can be cast in the form Σ��± = Σe ± Σ��12 , with

ΣAA/BBmn (z; xmn) =g2N ∑

k

zeikxmnz2 − !2

k, (4.53)

ΣABmn(z; xmn) =g2N ∑

k

!kei(kxmn−�k )z2 − !2

k, (4.54)

where xmn = xn − xm. It can be shown that

ΣBAmn(z; �, xmn) = ΣABnm(z; �, −xmn) = ΣABmn(z; −�, xmn − 1) . (4.55)

Again, these expressions in the thermodynamic limit can be evaluated substitutingthe sum by an integral, which can be computed easily with the change of variabley = exp(i sign(x12)k), giving the results shown in Eqs. (4.5) and (4.6).

4.B. QUANTUM OPTICAL MASTER EQUATION 85

4.B. �antum optical master eqationFrom Eq. (4.41), we can readily obtain the expression of HI in the interaction picture

HI (t) =∑nei!e t�neg ⊗ Bn(t) + H.c. , (4.56)

where

Bn(t) ={

g√2N ∑k ei(kxn+�k ) (e−i!k tuk + ei!k t lk) if n ∈ SA .g√2N ∑k eikxn (e−i!k tuk − ei!k t lk) if n ∈ SB .

(4.57)

Now, expanding the integrand in Eq. (2.18), neglecting the fast-rotating terms ∝ e±i2!e t ,we arrive at

�S(t) = −∑m, n

�meg�nge�S(t) ∫∞

0ds ei!es⟨Bm(t)B†n (t − s)⟩

+∑m, n

�nge�S(t)�meg ∫∞

0ds e−i!es⟨Bm(t − s)B†n (t)⟩

+∑m, n

�nge�S(t)�meg ∫∞


−∑m, n

�S(t)�meg�nge ∫∞

0ds e−i!es⟨Bm(t − s)B†n (t)⟩ . (4.58)

Let us compute the bath correlations assuming that the bath is in the vacuum state. Ifboth m, n ∈ SB,

∫∞


= g22N ∑

keik(xm−xn) ∫

∞

0ds ei!es (e−i!ks + ei!ks) (4.59)

= i g2

2N ∑keik(xm−xn)(

1!e + i0+ − !k

+ 1!e + i0+ + !k)

(4.60)

= iΣBBmn(!e + i0+) . (4.61)

In the last equality we have substituted the de�nition of the collective self energy forthe BB con�guration, Eq. (4.53). Similarly, the other correlator can be computed notingthat ⟨Bm(t)B†n (t′)⟩ only depends on the time di�erence t − t′, so the integral is the samechanging s → −s.

∫∞

0ds e−i!es⟨Bm(t − s)B†n (t)⟩ = −iΣBBmn(!e − i0+) . (4.62)


Analogously if m ∈ SA and n ∈ SB,

∫∞


= g22N ∑

kei[k(xm−xn)+�k ] ∫

∞

0ds ei!es (e−i!ks − ei!ks) (4.63)

= i g2

2N ∑kei[k(xm−xn)+�k ](

1!e + i0+ − !k

− 1!e + i0+ + !k)

(4.64)

= iΣABmn(!e + i0+) , (4.65)

and

∫∞

0ds e−i!es⟨Bm(t − s)B†n (t)⟩ = −iΣABmn(!e − i0+) . (4.66)

So in general, we can replace

∫∞

0ds ei!es⟨Bm(t)B†n (t − s)⟩ = iΣ��mn(!e + i0+) , (4.67)

∫∞

0ds e−i!es⟨Bm(t − s)B†n (t)⟩ = −iΣ��mn(!e − i0+) . (4.68)

Finally, splitting the self-energies in their real and imaginary parts, Σ��mn(!e ± i0+) =J ��mn ∓ iΓ��mn/2, gathering the terms that go with J ��mn and those that go with Γ��mn,

�S = −i∑m, n

J ��mn[�meg�nge , �S] +∑m, n

Γ��mn2 (2�nge�S�meg − �meg�nge�S − �S�meg�nge) . (4.69)

Back to the Schrödinger picture we have

�S = −i[HS , �S] − i∑m, n

J ��mn[�meg�nge , �S]

+ ∑m, n

Γ��mn2 (2�nge�S�meg − �meg�nge�S − �S�meg�nge) . (4.70)

4.C. Algebraic decayThe fractional decay of the emitter can be better seen when the emitter’s frequency isprecisely at any of the band edges. There, the contribution of the branch cuts on thedynamics is larger. De�ning

D(t) ≡ e(t) −∑zBSR(zBS)e−izBSt , (4.71)

4.C. ALGEBRAIC DECAY 87

at long times we have

limt→∞

D(t) ≃∑j BC,j(t) =∑

jKj(t)e−ixj t , (4.72)

with

Kj(t) =±12� ∫

∞

0dy 2Σe(xj − iy)e−yt

(xj − iy − !e)2 − Σ2e(xj − iy). (4.73)

The long-time average of the decaying part of the dynamics can be computed as

|D(t)|2 ≡ limt→∞

1t ∫

t

0dt′ |D(t′)|2 =∑

j|Kj(t)|2 . (4.74)

If the emitter’s transition frequency is close to one of the band edges, !e ≃ x0, then|D(t)|2 ≃ |K0(t)|2. In the long-time limit, we can expand the integrand of (4.73) in powerseries around y = 0,

K0(t) =±12� ∫

∞

0dy [

4g2

√i(2 − x20 + 2�2)

x0+ (y)] y

1/2e−yt (4.75)

≃ ±1√�g2

√i(2 − x20 + 2�2)

x0t−3/2 + (t−5/2) . (4.76)

Therefore, to leading order |D(t)|2 ∼ t−3.

5Conclusions and outlook/

Conclusiones y perspectiva

In this thesis we have studied problems that generalize the physics of topological insu-lators. In the �rst part, we analyze the dynamics of doublons in 1D and 2D topologicallattices. On the second part, we investigate the dynamics of quantum emitters interact-ing with a common topological waveguide QED bath, namely, a photonic analogue ofthe SSH model.

To understand the dynamics of doublons, we have derived an e�ective single-particleHamiltonian taking into account also the e�ect of a periodic driving. It contains twoterms: one corresponding to an e�ective doublon hopping renormalized by the driving,and another one corresponding to an e�ective on-site chemical potential. This helped usunderstand unusual phenomena that constrain doublon motion. For example, Shockley-like edge states can be induced in any �nite lattice by reducing the e�ective doublonhopping with the driving. These states may or may not compete against topologicaledge states depending on the dimensionality of the lattice. For 1D lattices, topologicalphases require the presence of chiral (sublattice) symmetry, which is spoiled by theon-site chemical potential. On the other hand, for 2D lattices threaded by a magnetic�ux no symmetries are required, and topological edge states coexist with Shockley-likeedge states. We demonstrate that edge states, either topological or not, can be used toproduce the transfer of doublons between distant sites (on the edge) of any �nite lattice.Furthermore, in 2D lattices with sites with di�erent number of neighbors, doublon’sdynamics can be con�ned to just one sublattice. We also analyze the feasibility ofdoublon experiments in noisy systems such as arrays of QDs, and estimate a doublonlifetime on the order of 10 ns for current devices.

For the analysis of quantum emitter dynamics, we have employed di�erent tech-niques valid in the Markovian and non-Markovian regimes. When the emitters arespectrally tuned to one of the band gaps, the non-trivial topology of the bath leads to theemergence of chiral photon bound states, which are localized on the left or right of theQE depending on the sign of the dimerization constant � . This gives rise to directionalinteractions between the emitters. Speci�cally, when the emitter frequency is tuned tothe middle of the inner band gap, the interaction between emitters coupled to di�erent

89

90 CHAPTER 5. CONCLUSIONS AND OUTLOOK

sublattices can be toggled on and o� by changing the sign of � . When the emitters arespectrally tuned to one of the bath’s bands, di�erent super/subradiant states appeardepending on the sign of � . This leads to di�erent behavior when a single photonscatters o� one QE coupled both to the A and B cavities in a single unit cell, or twoQEs coupled to di�erent sublattices. Last, we analyze the e�ective spin Hamiltoniansthat can be generated in the many-emitter case, and compute its phase diagram withexact diagonalization techniques. We �nd that for short-range interactions the emittersrealize a valence bond solid phase, while for long-range interactions a double Néelorder emerges.

One of the attractive points of our predictions is that they can be observed inseveral platforms by combining tools that, in most of the cases, have been alreadyimplemented experimentally. Regarding the �rst part, doublons have been observedin several experiments using cold atoms trapped in optical lattices [84, 87, 88]. Also,the SSH model has been realized in this kind of setups [31]. As for the second part,the photonic analogue of the SSH model has been implemented in several photonicplatforms [128–131], including some recent photonic crystal realizations [132]. Thelatter are particularly interesting due to the recent advances in their integration withsolid-state and natural atomic emitters (see Refs. [133, 134] and references therein).Superconducting metamaterials mimicking standard waveguide QED are now beingroutinely built and interfaced with one or many qubits in experiments [135, 136]. Theonly missing piece is the periodic modulation of the couplings between cavities to obtainthe SSH model, for which there are already proposals using circuit superlattices [137].Furthermore, Quantum optical phenomena can be simulated in pure atomic scenariosby using state-dependent optical lattices. The idea is to have two di�erent trappingpotentials for two atomic metastable states, such that one state mostly localizes, playingthe role of QEs, while the other state propagates as a matter-wave. This proposal [138]has been recently used [139] to explore the physics of standard waveguide baths. Beyondthese platforms, the bosonic analogue of the SSH model has also been discussed in thecontext of metamaterials [140] or plasmonic and dielectric nanoparticles [141, 142],where the predicted phenomena could as well be observed.

Topological matter is a very active research �eld in which important advances,both on theoretical and applied grounds, have been produced in recent years. Theresearch here presented demonstrates the variety of phenomena that appear at thecrossover between this and other �elds of physics. This is a rather new and unexploredresearch direction, which surely will provide exciting discoveries in the near future.Prospective studies could investigate the dynamics of doublons in 1D lattices withhigher topological invariants [2*], or the use of topological edge states to transferfew-particle states other than doublons between distant regions of a lattice. It wouldalso be interesting to analyze the dynamics of several doublons, or the dynamics ofthe Hubbard model in the intermediate interaction regime, where the interaction is of

91

the order of the hopping. Regarding the dynamics of quantum emitters, it would beinteresting to study other topological baths. For example, the phenomena associated tothe SSH bath would have analogues in higher dimensions considering photonic bathsafter higher-order topological insulators [143, 144]. As for the photonic SSH bath, weare currently working on an in-depth survey of the many-body phases that appear ineach of the band gaps. Also considering other types of emitters (with a more complexlevel structure) which would allow for di�erent e�ective spin interactions [118, 119]./

En esta tesis hemos estudiado problemas que generalizan la física de los aislantes topo-lógicos. En la primera parte analizamos la dinámica de dublones en redes topológicas1D y 2D. En la segunda parte, investigamos la dinámica de emisores cuánticos queinteractúan con un baño común topológico tipo guía de ondas, concretamente, con unanálogo fotónico del modelo SSH.

Para entender la dinámica de los dublones, hemos derivado un Hamiltoniano efec-tivo de una partícula que además incluye el efecto de una modulación periódica delsistema (driving en inglés). Este Hamiltoniano efectivo contiene dos términos: unose corresponde con el salto de dublones en la red, renormalizado por el driving, y elotro se corresponde con un potencial químico local efectivo. Esto nos ha permitidoentender fenómenos inusuales que constriñen la dinámica de los dublones. Por ejemplo,estados de borde de tipo Shockley pueden inducirse en cualquier red �nita reducien-do el salto del dublón mediante el driving. Estos estados pueden competir o no conestados de borde topológicos, en función de la dimensión de la red. En redes 1D, lasfases topológicas requieren la presencia de simetría quiral (simetría de subred), la cualse rompe debido al potencial químico local. Por otro lado, en redes 2D atravesadaspor un �ujo de campo magnético, no se requiere ninguna simetría para tener fasestopológicas, y los estados de borde topológicos pueden coexistir con estados de borde detipo Shockley. Demostramos que los estados de borde, ya sean topológicos o no, puedenusarse para transferir dublones entre sitios distantes (en el borde) de cualquier red �nita.Además, en redes 2D con sitios con distinto índice de coordinación, la dinámica de losdublones puede con�narse a una única subred. También analizamos la posibilidad dehacer experimentos con dublones en sistemas ruidosos como son las cadenas de puntoscuánticos, y estimamos para el dublón una vida media del orden de 10 ns en dispositivosactuales.

Para el análisis de la dinámica de emisores cuánticos, hemos empleado distintastécnicas válidas en el régimen Markoviano y no-Markoviano. Cuando la frecuenciade los emisores se encuentra en uno de los band gaps, la topología no trivial del bañoproduce la aparición de estados ligados de fotones que son quirales, es decir, que estánlocalizados a la izquierda o derecha del emisor en función del signo de la constantede dimerización � . Esto da lugar a interacciones direccionales entre los emisores. Enconcreto, cuando la frecuencia de los emisores está ajustada al centro del band gap

92 CHAPTER 5. CONCLUSIONS AND OUTLOOK

interno, la interacción entre emisores acoplados a subredes distintas puede activarse ydesactivarse cambiando el signo de � . Cuando la frecuencia de los emisores se encuentraen una de las bandas del baño, distintos estados super/subradiantes aparecen en funcióndel signo de � . Esto produce un comportamiento distinto en función de la topologíadel baño cuando un foton se dispersa a través de dos emisores acoplados a redesdistintas. Por último, analizamos el Hamiltoniano de spin que puede generarse enel caso de muchos emisores, y calculamos su diagrama de fases mediante técnicasde diagonalización exacta. Encontramos que para interacciones de corto alcance losemisores realizan un sólido de enlaces de valencia, mientras que para interacciones delargo alcance aparece un orden de tipo Néel doble.

Uno de los puntos atractivos de nuestras predicciones es que pueden observarseen varias plataformas combinando herramientas que, en la mayoría de los casos, yahan sido implementadas experimentalmente. Respecto a la primera parte, los dubloneshan sido observados en varios experimentos utilizando átomos ultrafríos atrapados enredes ópticas [84, 87, 88]. Además, el modelo SSH ya ha sido realizado en este tipo deexperimentos [31]. Respecto a la segunda parte, el análogo del modelo SSH ha sidoimplementado en varias plataformas fotónicas [128-131], incluidas algunas realizacionesde cristales fotónicos [132] que son particularmente interesantes debido a la recienteintegración de emisores naturales y de estado sólido en las mismas (ver Refs. [133, 134]y referencias allí mencionadas). Metamateriales superconductores que imitan guíasde onda cuánticas se construyen ahora de forma rutinaria y ya hay experimentos enlos que se acoplan con uno o varios qubits [135, 136]. La única pieza que falta es lamodulación periódica de los acoplos entre cavidades para obtener el modelo SSH, para locual ya hay propuestas utilizando superredes de circuitos [137]. Asimismo, fenómenosde la óptica cuántica pueden simularse en sistemas púramente atómicos utilizandoredes ópticas dependientes de los estados cuánticos de los átomos. La idea es tener dospotenciales distintos para dos estados atómicos metaestables, de forma que un estadoestá mayormente localizado, jugando el papel de los emisores, mientras que el otroestado se propaga como una onda de materia. Esta propuesta [138] ha sido utilizadarecientemente [139] para explorar la física de baños de guía de ondas estándar. Más alláde estas plataformas, el análogo bosónico del modelos SSH también se ha discutido enel contexto de los metamateriales [140] o de sistemas plasmónicos [141, 142], donde losfenómenos predichos podrían observarse también.

La materia topológica es un campo de investigación muy activo en el que se han pro-ducido importantes avances tanto a nivel teórico como práctico en los últimos años. Lasinvestigaciones aquí presentadas demuestran la variedad de fenómenos que aparecen alcombinar este campo con otros campos de la física. Esta es una dirección de investiga-ción aún nueva e inexplorada que seguramente dará lugar a grandes descubrimientos enun futuro cercano. Estudios futuros podrían investigar la dinámica de dublones en redes1D con invariantes topológicos más altos [2*], o el uso de estados de borde topológicos

93

para la transferencia de estados de pocas partículas distintos de los dublones entreregiones distantes de una red. También sería interesante analizar la dinámica de variosdublones, o la dinámica del modelo de Hubbard en el régimen de interacción intermedio,en el que ésta es del mismo orden que el salto de las partículas. Respecto a la dinámica deemisores cuánticos, sería interesante estudiar otros baños topológicos. Por ejemplo, losfenómenos descritos para el baño tipo SSH tendrían análogos en dimensiones mayoresconsiderando baños similares a aislantes topológicos de orden más alto [143, 144]. Encuanto al baño fotónico SSH, estamos en estos momentos realizando un análisis enprofundidad de las distintas fases que aparecen en cada uno de los band gaps. Además,estamos considerando también otros tipos de emisores (con una estructura de nivelesmás compleja) que permitirían generar distintas interacciones de spin efectivas [118,119].

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