COOPERATIVE EFFECTS IN QUANTUM SYSTEMS: SUPERRADIANCE AND LONG-RANGE INTERACTIONS

Questa tesi di dottorato studia l’interazione della cooperatività con il rumore in sistemi realistici, focalizzandosi principalmente sulla superradianza. Gli effetti cooperativi emergono dall’interazione collettiva di un insieme di elementi con un campo esterno. Esempi degni di nota sono la superconduttività, dove le coppie di Cooper elettroniche interagiscono con le vibrazioni reticolari, le eccitazioni di plasma, che sorgono dall'interazione collettiva degli elettroni in un metallo con il campo coulombiano, e la superradianza, ovvero quel processo di emissione spontanea cooperativa che sorge da un aggregato di emettitori identici. Gli effetti cooperativi sono tipicamente robusti al disordine e al rumore, cosa che li rende interessanti per delle applicazioni a dispositivi quantistici che possano operare a temperatura ambiente. In questo lavoro, inizialmente, introduciamo un formalismo di “master equations” che descrive l’accoppiamento collettivo di un aggregato di emettitori/assorbitori con il campo elettromagnetico, valido quando le dimensioni dell'aggregato sono sia maggiori che minori della lunghezza d’onda emessa/assorbita. Inoltre, il formalismo è valido per accoppiamento sia debole che forte con il campo elettromagnetico e, cosa più importante, permette di descrivere correttamente la superradianza in diversi regimi. In tale formalismo, studiamo l’interazione tra superradianza e rumore termico sia per nanotubi molecolari (di dimensioni minori della lunghezza d’onda associata alla transizione) che sono presenti nei complessi antenna fotosintetici dei Green Sulfur Bacteria, sia pure per superreticoli di quantum dots di nuova generazione, aventi dimensioni maggiori della lunghezza d’onda emessa. In entrambi i casi si dimostra che la coerenza può permanere in presenza di rumore termico alle temperature a cui questi sistemi sono stati analizzati sperimentalmente (temperatura ambiente per i nanotubi molecolari, e 6 K per i superreticoli di quantum dots). Nello specifico, nei nanotubi molecolari mostriamo che la macroscopica delocalizzazione coerente delle eccitazioni a temperatura ambiente, che copre centinaia di molecole, può essere considerata un effetto emergente che origina dall’effetto combinato della specifica disposizione geometrica delle molecole e della presenza di accoppiamenti tra subunità del cilindro, incrementati dagli effetti cooperativi. Questi risultati aprono la strada a nuovi modi per ingegnerizzare dei “quantum wires” robusti al rumore grazie alla cooperatività. Inoltre, la presente analisi di sistemi allo stato solido basati su superreticoli di “quantum dots” di perovskite (CsPbBr3) fornisce una base teorica in grado di comprendere recenti osservazioni di emissione superradiante. Sulla base della nostra teoria, suggeriamo che futuri esperimenti dove si utilizzino quantum dots più piccoli, potrebbe aumentare significativamente la robustezza del sistema al rumore termico, aprendo la strada verso la superradianza a temperatura ambiente in sistemi allo stato solido. Si considerano anche i complessi antenna dei Purple Bacteria, dove è ben risaputo che gli effetti cooperativi incrementano il trasferimento e l’accumulo di eccitazioni generate dalla luce assorbita. Mostriamo come queste proprietà possono essere sfruttate per creare un laser ispirato a sistemi biologici e basato su aggregati molecolari, dove la luce solare, benché debole, sarebbe utilizzata come sorgente di pompaggio. Il trasferimento efficiente di energia dentro questo sistema, all’atto pratico, focalizzerebbe l’eccitazione assorbita in direzione di un dimero molecolare, composto da una coppia di molecole interagenti, opportunamente scelte. L’orientazione dei momenti di dipolo di transizione in ciascun dimero è tale da concentrare tutta l’intensità del dipolo nel livello a più alta energia, lasciando lo stato eccitonico inferiore otticamente inattivo. Un dimero molecolare in tale configurazione, che è ideale per ottenere inversione di popolazione, è chiamato “H-dimer”. Tale H-dimer, nell’archittettura qui proposta per un laser ispirato a sistemi biologici, è posto al centro di un aggregato molecolare ispirato a sistemi biologici. Gli H-dimers, eccitati dagli aggregati molecolari circostanti, raggiungono inversione di popolazione e, dunque, possono emettere luce laser quando tali aggregati sono posti in una cavità ottica. Convertire l’energia incoerente fornita dal Sole in un fascio laser coerente supererebbe diverse limitazioni pratiche inerenti all’utilzzo della luce solare come sorgente di energia pulita. Per esempio, i fasci laser sono molto efficienti nell’avviare reazioni chimiche che convertono la luce solare in energia chimica. Inoltre, dal momento che i complessi fotosintetici batterici tendono ad operare nella regione spettrale del vicino infrarosso, la nostra proposta si presta in modo naturale a realizzare laser a infrarossi a corta lunghezza d’onda, i cui fasci viaggerebbero per lunghe distanze senza quasi perdere energia, quindi distribuendo in modo efficiente l’energia solare raccolta. Nella ricerca di un meccanismo comune alla cooperatività e alla sua robustezza, abbiamo confrontato il modello delle coppie di Cooper della superconduttività con la superradianza in singola eccitazione, mostrando molte somiglianze tra i due fenomeni: in particolare, i sistemi superradianti presentano una “gap” immaginaria nel piano complesso (ovvero, una segregazione tra i tempi di vita degli autostati del sistema) che, in modo simile alla gap superconduttiva, rende questi sistemi robusti al rumore statico. Più in generale, mostriamo che ogni interazione a lungo raggio tra i costituenti di un sistema induce effetti collettivi, manifestati da delle gap nello spettro eccitonico. Perciò, la nostra analisi successiva considera l’effetto delle interazioni a lungo raggio sul trasporto eccitonico lungo catene disordinate. Dimostriamo che la presenza di uno stato collettivo ben separato dagli altri stati influenza tutto lo spettro del sistema, generando dei regimi molto controintuitivi dove il trasporto è incrementato dal disordine o è indipendente da esso, e tali regimi si estendono su molti ordini di grandezza nell’intensità del disordine. Dimostriamo anche che una catena fortemente accoppiata a un modo del campo elettromagnetico in una cavità ottica è equivalente a una catena con interazione a lungo raggio, mostrandosi dunque molto promettente per esperimenti e applicazioni future. Nello specifico, mostriamo che catene molecolari realistiche, ioni intrappolati realizzati allo stato dell’arte e atomi di Rydberg sono tutti in grado di raggiungere l’intensità di interazione a lungo raggio tale per cui il trasporto sarebbe incrementato dal disordine o indipendente da esso, puntando alla realizzazione di un trasporto di energia senza dissipazione in “quantum wires” disordinati.This Ph.D. thesis studies the interplay of cooperativity and noise in realistic systems, largely focusing on superradiance. Cooperative effects emerge from the collective interaction of an ensemble of elements to an external field. Notable examples are superconductivity, where the electron Cooper pairs interact with the lattice vibrations, plasmon excitations, arising from the collective interaction of electrons in a metal with the Coulomb field, and superradiance, that is a cooperative spontaneous emission process stemming from an aggregate of identical emitters. Cooperative effects are typically robust to disorder and noise, making them interesting for applications to quantum devices operating at room temperature. In this work, we first present a general master equation formalism that describes the collective coupling of an aggregate of emitters/absorbers to the electromagnetic field, valid both when the size of the aggregate is larger or smaller than the emitted/absorbed wavelength. Also, the formalism is valid both for weak and strong coupling of the emitters to the electromagnetic field and, most importantly, it allows to correctly describe superradiance in different regimes. Within such formalism, the interplay of superradiance and thermal noise is studied both for molecular nanotubes (of size smaller than the transition wavelength) that are present in the antenna complexes of photosynthetic Green Sulfur Bacteria, and also for novel solid state quantum dot superlattices, having size larger than the emitted wavelength. In both cases it is shown that coherence can persist in presence of thermal noise at the temperatures where these systems have been experimentally analyzed (room temperature for molecular nanotubes, and 6 K for quantum dot superlattices). Specifically, in natural molecular nanotubes we show that the macroscopic coherent delocalization of the excitation at room temperature, covering hundreds of molecules, can be considered an emergent effect originating from the combined effect of the specific geometric disposition of the molecules and the presence of cooperatively enhanced couplings between cylinder subunits. These results open the path to new ways of engineering quantum wires robust to noise thanks to cooperativity. Moreover, our analysis of solid state systems based on perovskite (CsPbBr3) quantum dot superlattices provides a theoretical framework able to explain recent observations of superradiant emission. Based on our theory, we suggest that further experiments, using smaller quantum dots, could significantly increase the robustness of the system to thermal noise, paving the way towards room-temperature superradiance in solid-state systems. We also considered the antenna complexes of Purple Bacteria, where cooperative effects are well known to boost the transfer and storage of photo-absorbed excitations. We show how these properties can be exploited to create a bio-inspired molecular aggregate laser medium, where natural sunlight, although weak, would be used as a pumping source. The efficient energy transfer within this system would effectively focus the absorbed excitation on a suitably chosen molecular dimer, composed by a pair of interacting molecules. The orientation of the molecule transition dipole moment in each dimer is such to concentrate all the dipole strength in the highest energy level, leaving the lower excitonic state dark. A molecular dimer in such configuration, which is ideal to achieve population inversion, is called H-dimer. Such an H-dimer in our proposed architecture for a bio-inspired laser medium, is placed at the center of the bio-inspired molecular aggregates. The H-dimers, pumped by the surrounding molecular aggregates, reach population inversion and, therefore, can lase when such aggregates are placed in an optical cavity. Turning the incoherent energy supply provided by the Sun into a coherent laser beam would overcome several of the practical limitations inherent in using sunlight as a source of clean energy. For example, laser beams are highly effective at driving chemical reactions which convert sunlight into chemical energy. Further, since bacterial photosynthetic complexes tend to operate in the near-infrared spectral region, our proposal naturally lends itself for realising short-wavelength infrared lasers which would allow their beams to travel nearly losslessly over large distances, thus efficiently distributing the collected sunlight energy. In search of a common mechanism to cooperativity and its robustness, we have compared the Cooper pair model of superconductivity and single-excitation superradiance, showing many similarities between the two: in particular, superradiant systems present an imaginary gap in the complex plane (that is, a segregation between the lifetimes of the system eigenstates) that, similarly to the superconducting gap, makes these systems robust to static disorder. More in general, we show that any long-range interaction between the constituents of a system generates collective behaviours, manifested by gaps in the excitonic spectrum. Therefore, our further analysis considers the effect of long-range interactions on excitation transport along disordered chains. We show that the presence of a gapped, collective state affects the whole spectrum of the system, generating quite counter-intuitive disorder-enhanced and disorder-independent transport regimes, that extend over many orders of magnitude of the disorder strength. We also prove that a chain strongly coupled to a cavity mode is equivalent to a long-range interacting chain, thus being very promising for future experiments and applications. Specifically, we show that realistic molecular chains, state-of-the-art trapped ions and Rydberg atoms are all able to reach the needed long-range interaction strength that would show disorder-enhanced or disorder-independent transport, aiming to the realization of dissipationless transport of energy in disordered quantum wires

Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks: validity and effectiveness

We investigate the validity of the non-Hermitian Hamiltonian approach in describing quantum transport in disordered tight-binding networks connected to external environments, acting as sinks. Usually, non-Hermitian terms are added, on a phenomenological basis, to such networks to summarize the effects of the coupling to the sinks. Here we consider a paradigmatic model of open quantum network for which we derive a non-Hermitian effective model, discussing its limit of validity by a comparison with the analysis of the full Hermitian model. Specifically, we consider a ring of sites connected to a central one-dimensional lead. The lead acts as a sink which absorbs the excitation initially present in the ring. The coupling strength to the lead controls the opening of the ring system. This model has been widely discussed in literature in the context of light-harvesting systems. We analyze the effectiveness of the non-Hermitian description both in absence and in presence of static disorder on the ring. In both cases, the non-Hermitian model is valid when the energy range determined by the eigenvalues of the ring Hamiltonian is smaller than the energy band in the lead. Under such condition, we show that results about the interplay of opening and disorder, previously obtained within the non-Hermitian Hamiltonian approach, remain valid when the full Hermitian model in presence of disorder is considered. The results of our analysis can be extended to generic networks with sinks, leading to the conclusion that the non-Hermitian approach is valid when the energy dependence of the coupling to the external environments is sufficiently smooth in the energy range spanned by the eigenstates of the network.Comment: Final version, 20 pages, 15 figure

Enhanced robustness and dimensional crossover of superradiance in cuboidal nanocrystal superlattices

Cooperative emission of coherent radiation from multiple emitters (known as superradiance) has been predicted and observed in various physical systems, most recently in CsPbBr$_3$ nanocrystal superlattices. Superradiant emission is coherent and occurs on timescales faster than the emission from isolated nanocrystals. Theory predicts cooperative emission being faster by a factor of up to the number of nanocrystals ($N$). However, superradiance is strongly suppressed due to the presence of energetic disorder, stemming from nanocrystal size variations and thermal decoherence. Here, we analyze superradiance from superlattices of different dimensionalities (1D, 2D and 3D) with variable nanocrystal aspect ratios. We predict as much as a thirty-fold enhancement in robustness against realistic values of energetic disorder in three-dimensional (3D) superlattices composed of cuboid-shaped, as opposed to cube-shaped, nanocrystals. Superradiance from small $(N\lesssim 10^3)$ two-dimensional (2D) superlattices is up to 10 times more robust to static disorder and up to twice as robust to thermal decoherence than three-dimensional (3D) superlattices with the same $N$. As the number of $N$ increases, a crossover in the robustness of superradiance occurs from 2D to 3D superlattices. For large $N\ (> 10^3)$, the robustness in 3D superlattices increases with $N$, showing cooperative robustness to disorder. This opens the possibility of observing superradiance even at room temperature in large 3D superlattices, if nanocrystal size fluctuations can be kept small

Scale-invariant phase transition of disordered bosons in one dimension

The disorder-induced quantum phase transition between superfluid and non-superfluid states of bosonic particles in one dimension is generally expected to be of the Berezinskii-Kosterlitz-Thouless (BKT) type. Here, we show that hard-core lattice bosons with integrable power-law hopping decaying with distance as $1/r^\alpha$ - corresponding in spin language to a $XY$ model with power-law couplings - undergo a non-BKT continuous phase transition instead. We use exact quantum Monte-Carlo methods to determine the phase diagram for different values of the exponent $\alpha$, focusing on the regime $\alpha > 2$. We find that the scaling of the superfluid stiffness with the system size is scale-invariant at the transition point for any $\alpha\leq 3$ - a behavior incompatible with the BKT scenario and typical of continuous phase transitions in higher dimension. By scaling analysis near the transition point, we find that our data are consistent with a correlation length exponent satisfying the Harris bound $\nu \geq 2$ and demonstrate a new universal behavior of disordered bosons in one dimension. For $\alpha>3$ our data are consistent with a BKT scenario where the liquid is pinned by infinitesimal disorder

Anomalous diffusion in the Long-Range Haken-Strobl-Reineker model

We analyze the propagation of excitons in a $d$-dimensional lattice with power-law hopping $\propto 1/r^\alpha$ in the presence of dephasing, described by a generalized Haken-Strobl-Reineker model. We show that in the strong dephasing (quantum Zeno) regime the dynamics is described by a classical master equation for an exclusion process with long jumps. In this limit, we analytically compute the spatial distribution, whose shape changes at a critical value of the decay exponent $\alpha_{\rm cr} = (d+2)/2$. The exciton always diffuses anomalously: a superdiffusive motion is associated to a L\'evy stable distribution with long-range algebraic tails for $\alpha\leq\alpha_{\rm cr}$, while for $\alpha > \alpha_{\rm cr}$ the distribution corresponds to a surprising mixed Gaussian profile with long-range algebraic tails, leading to the coexistence of short-range diffusion and long-range L\'evy-flights. In the many-exciton case, we demonstrate that, starting from a domain-wall exciton profile, algebraic tails appear in the distributions for any $\alpha$, which affects thermalization: the longer the hopping range, the faster equilibrium is reached. Our results are directly relevant to experiments with cold trapped ions, Rydberg atoms and supramolecular dye aggregates. They provide a way to realize an exclusion process with long jumps experimentally.Comment: 5 pages, 2 figure

Engineering random spin models with atoms in a high-finesse cavity

All-to-all interacting, disordered quantum many-body models have a wide range of applications across disciplines, from spin glasses in condensed-matter physics, over holographic duality in high-energy physics, to annealing algorithms in quantum computing. Typically, these models are abstractions that do not find unambiguous physical realisations in nature. Here, we realise an all-to-all interacting, disordered spin system by subjecting an atomic cloud in a cavity to a controllable light shift. Adjusting the detuning between atom resonance and cavity mode, we can tune between disordered versions of a central-mode model and a Lipkin-Meshkov-Glick model. By spectroscopically probing the low-energy excitations of the system, we explore the competition of interactions with disorder across a broad parameter range. We show how disorder in the central-mode model breaks the strong collective coupling, making the dark state manifold cross over to a random distribution of weakly-mixed light-matter, "grey", states. In the Lipkin-Meshkov-Glick model the ferromagnetic finite-size ground state evolves towards a paramagnet as disorder is increased. In that regime, semi-localised eigenstates emerge, as we observe by extracting bounds on the participation ratio. These results present significant steps towards freely programmable cavity-mediated interactions for the design of arbitrary spin Hamiltonians.Comment: 8 pages, 4 figures, methods, supplementary materia

Macroscopic coherence as an emergent property in molecular nanotubes

Nanotubular molecular self-aggregates are characterized by a high degree of symmetry and they are fundamental systems for light-harvesting and energy transport. While coherent effects are thought to be at the basis of their high efficiency, the relationship between structure, coherence and functionality is still an open problem. We analyse natural nanotubes present in Green Sulphur Bacteria. We show that they have the ability to support macroscopic coherent states, i.e. delocalized excitonic states coherently spread over many molecules, even at room temperature. Specifically, assuming a canonical thermal state we find, in natural structures, a large thermal coherence length, of the order of 1000 molecules. By comparing natural structures with other mathematical models, we show that this macroscopic coherence cannot be explained either by the magnitude of the nearest-neighbour coupling between the molecules, which would induce a thermal coherence length of the order of 10 molecules, nor by the presence of long-range interactions between the molecules. Indeed we prove that the existence of macroscopic coherent states is an emergent property of such structures due to the interplay between geometry and cooperativity (superradiance and super-transfer). In order to prove that, we give evidence that the lowest part of the spectrum of natural systems is determined by a cooperatively enhanced coupling (super-transfer) between the eigenstates of modular sub-units of the whole structure. Due to this enhanced coupling strength, the density of states is lowered close to the ground state, thus boosting the thermal coherence length. As a striking consequence of the lower density of states, an energy gap between the excitonic ground state and the first excited state emerges. Such energy gap increases with the length of the nanotube (instead of decreasing as one would expect), up to a critical system size which is close to the length of the natural complexes considered

Real and imaginary energy gaps: a comparison between single excitation Superradiance and Superconductivity and robustness to disorder

A comparison between the single particle spectrum of the discrete Bardeen-Cooper-Schrieffer (BCS) model, used for small superconducting grains, and the spectrum of a paradigmatic model of Single Excitation Superradiance (SES) is presented. They are both characterized by an equally spaced energy spectrum (Picket Fence) where all the levels are coupled between each other by a constant coupling which is real for the BCS model and purely imaginary for the SES model. While the former corresponds to the discrete BCS-model describing the coupling of Cooper pairs in momentum space and it induces a Superconductive regime, the latter describes the coupling of single particle energy levels to a common decay channel and it induces a Superradiant transition. We show that the transition to a Superradiant regime can be connected to the emergence of an imaginary energy gap, similarly to the transition to a Superconductive regime where a real energy gap emerges. Despite their different physical origin, it is possible to show that both the Superradiant and the Superconducting gaps have the same magnitude in the large gap limit. Nevertheless, some differences appear: while the critical coupling at which the Superradiant gap appears is independent of the system size N, for the Superconductivity gap it scales as (ln N)−1, which is the expected BCS result. The presence of a gap in the imaginary energy axis between the Superradiant and the Subradiant states shares many similarities with the “standard” gap on the real energy axis: the superradiant state is protected against disorder from the imaginary gap as well as the superconducting ground state is protected by the real energy gap. Moreover we connect the origin of the gapped phase to the long-range nature of the coupling between the energy levels

Bio-inspired natural sunlight-pumped lasers

Even though sunlight is by far the most abundant renewable energy source available to humanity, its dilute and variable nature has kept efficient ways to collect, store, and distribute this energy tantalisingly out of reach. Turning the incoherent energy supply provided by the Sun into a coherent laser beam would overcome several of the practical limitations inherent in using sunlight as a source of clean energy: laser beams travel nearly losslessly over large distances, and they are effective at driving chemical reactions which convert sunlight into chemical energy. Here we propose a bio-inspired blueprint for a novel type of laser with the aim of upgrading unconcentrated natural sunlight into a coherent laser beam. Our proposed design constitutes a novel and different path towards sunlight-pumped lasers. In order to achieve lasing with the extremely dilute power provided by natural sunlight, we here propose a laser medium comprised of molecular aggregates inspired by the architecture of natural photosynthetic complexes. Such complexes exhibit a very large internal efficiency in harvesting photons from a power source as dilute as natural sunlight. Specifically, we consider a hybrid structure, where photosynthetic complexes in purple bacteria (Rhodobacter sphaeroides) surround a suitably engineered molecular dimer composed of two strongly coupled chromophores. We show that if pumped by the surrounding photosynthetic complex, which efficiently collects and concentrates solar energy, the core dimer structure can reach population inversion, and reach the lasing threshold under natural sunlight. The design principles proposed here will also pave the way for developing other bio-inspired quantum devices.Comment: Main text: 24 pages, 6 figures. Supplementary Material: 25 pages, 9 figure