Anomalous and Critical Dynamics in a Dissipative Central Spin System
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- Jan Schulze
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1 DISS. ETH NO Anomalous and Critical Dynamics in a Dissipative Central Spin System A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by MENA ELISABETH ISSLER MSc ETH in Physics born on citizen of Zurich and the United States of America accepted on the recommendation of Prof. Dr. Ataç Imamoğlu Prof. Dr. Iacopo Carusotto 2015
2 Abstract The phenomena of anomalous and critical dynamics occur across a wide range of systems. Anomalous dynamics is a diffusion process that is characterized by rare events, termed Lévy flights. Critical dynamics is associated with diverging system properties in the vicinity of a critical point at which the system undergoes a phase transition. These effects can be studied not only in classical systems but also in quantum systems. A quantum system of particular interest is the central spin system, since several candidate systems for the building blocks (qubits) of quantum information processing applications can be mapped onto it. Here, a single central spin interacts with a large spin environment. Understanding the intricate coupled dynamics is key in overcoming a major roadblock to quantum information applications, which is rooted in the dissipation and decoherence of a spin qubit. Quantum dots, which are zero-dimensional nanostructures that confine electrons in all three spatial dimensions, constitute a qubit candidate. The confined electron spin interacts with the nuclear spin environment of the host material, realizing a driven-dissipative central spin system. Since many parameters of a quantum dot can be tuned, it is an extremely versatile system to study both experimentally and theoretically. In this context, we identify open challenges, which include the preparation of the spin environment of a spin qubit in an ultra-narrow state to reduce the spin decoherence, the direct observation of Lévy flights in a single quantum system, as well as the investigation of critical dynamics of a dissipative solid-state system in an experimentally accessible regime. We address these questions theoretically in a driven-dissipative central spin system with a particular focus on an optically driven self-assembled quantum dot. When the central (electron) spin is driven by two lasers in a coherent population trapping configuration, the coupling to the (nuclear) spin environment forces the system into an electron nuclei dark state. As a consequence, the optical absorption is drastically reduced, which in turn suppresses the dominant nuclear spin dynamics. The standard deviation of the resulting ultra-narrow nuclear spin distribution is close to the single-spin regime. This greatly prolongs the spin coherence time of the electron, which is an important result in light of potential applications in quantum information processing. Remarkably, the mechanism that leads to the cooling of the nuclear spin environment can be described in terms of an anomalous diffusion process of the nuclear spins. We then show that anomalous nuclear diffusion can also be observed in a time-dependent scheme where the electron is driven by periodic laser pulses. Strikingly, these nuclear Lévy flights can be monitored in real-time by tracking the photon emission events, making this scheme a promising path towards the experimental observation of Lévy flights in a single quantum system. Finally, we find signatures reminiscent of critical phenomena in a spin environment that is controlled by the central spin. The latter is assumed to be driven by two lasers in the coherent population trapping configuration. For selfassembled quantum dots, the nuclear spins are forced to transition continuously from an unpolarized to a polarized state when the relative strength of the two lasers is tuned accordingly. The transition is accompanied by increased nuclear fluctuations. However, in the thermodynamic limit the feedback from the electron i
3 Abstract to the nuclear spins diminishes as a result of the asymmetric coupling between the single electron and the large nuclear environment. Nevertheless, assuming a constant coupling strength, we observe that the spectral gap at the critical point closes in the thermodynamic limit, providing evidence of a non-analytic change of the steady state at the critical point. ii
4 Zusammenfassung Die Phänomene der anomalen und kritischen Dynamik treten in einer Vielzahl von Systemen auf. Anomale Dynamik ist ein Diffusionsprozess, welcher durch seltene Ereignisse bestimmt wird, wobei hierfür auch der Begriff Lévy flights verwendet wird. Kritische Dynamik tritt in Verbindung mit divergierenden Systemeigenschaften in der Umgebung eines kritischen Punktes eines Phasenüberganges auf. Diese Effekte können nicht nur in klassischen, sondern auch quantenmechanischen Systemen untersucht werden. Das central spin model ist ein quantenmechanisches System, in dem ein einzelner central Spin mit seiner Umgebung, welche aus einer grossen Anzahl Spins besteht, wechselwirkt. Verschiedene Systeme, welche potenziell als Grundbausteine (Qubits) für die Quanteninformationsverarbeitung dienen könnten, werden durch das central spin model beschrieben. Um eine der grössten Herausforderungen auf dem Weg zu Quanteninformationsanwendungen - die Dissipation und Dekohärenz des Spin-Qubits - zu überwinden, ist es von wesentlicher Bedeutung die komplexe Dynamik dieses gekoppelten Systems zu verstehen. Eine potenzielle Realisation eines solchen Qubits ist ein Quantenpunkt. Dieser ist eine null-dimensionale Nanostruktur, welche Elektronen in allen drei Raumdimensionen einschliesst. Das gefangene Elektron wechselwirkt mit den Kernspins im umgebenden Material und stellt ein getriebenes und dissipatives central spin model dar. Da viele Parameter eines Quantenpunkts angepasst und gemessen werden können, ist dies ein extrem vielseitiges System für experimentelle und theoretische Untersuchungen. In dieser Arbeit werden Herausforderungen im Zusammenhang mit dem getriebenen und dissipativen central spin model theoretisch besprochen und an selbstorganisierten Quantenpunkten illustriert. Darunter fallen: Erstens, die Herstellung eines Zustandes der Spin-Umgebung eines Spin-Qubits mit einer ultra-schmalen Verteilung mit dem Ziel die Dekohärenz des Qubits zu verringern. Zweitens, die direkte Beobachtung von Lévy flights in einem einzelnen Quantensystem. Drittens, die Untersuchung der kritischen Dynamik in einem dissipativen Festkörpersystem, welches sich in einem experimentell realisierbaren Regime befindet. Wenn ein Elektronspin mit zwei Lasern in der sogenannten coherent population trapping Konfiguration angeregt wird, führt die Kopplung an die Kernspi- numgebung dazu, dass das Elektron-Kernspin-System in einen dunklen Zustand überführt wird. Infolgedessen nimmt die optische Absorption drastisch ab, wodurch die dominante Kernspindynamik unterdrückt wird. Die Standardabweichung der resultierenden ultra-schmalen Kernspinverteilung ist nahe am Einzelnen-Spin Regime. Dies verlängert die Dekohärenzzeit des Elektrons stark und stellt damit in Anbetracht möglicher Anwendungen in der Quanteninformationsverarbeitung ein wichtiges Resultat dar. Erstaunlicherweise kann der Mechanismus, welcher zu der Kühlung der Kernspins führt, als anomale Diffusion beschrieben werden. Ferner zeigen wir, dass die anomale Diffusion der Kernspins auch in einem zeitabhängigen Messschema beobachtet werden kann, wobei das Elektron hier mit einem periodischen Laserpuls angeregt wird. Bemerkenswerterweise können die Lévy flights der Kernspins durch das Beobachten der Photonemissionen in Echtzeit verfolgt werden. Deshalb stellt dieses Messschema einen erfolgversprechenden Weg zur experimentellen Beobachtung von Lévy flights in einem einzelnen Quantensys- iii
5 Zusammenfassung tem dar. Schliesslich beobachten wir Anzeichen kritischer Phänomene in einer Spin-Umgebung (Kernspinumgebung), deren Dynamik von einem central Spin (Elektron) bestimmt wird. Hierbei wird angenommen, dass das Elektron von zwei Lasern in einer coherent population trapping Konfiguration angeregt wird. Die Kernspins eines selbstorganisierten Quantenpunkts gehen von einem unpolarisierten in einen polarisierten Zustand über, wenn die relative Stärke beider Laser entsprechend eingestellt wird. Dieser Übergang geht einher mit zunehmenden Fluktuationen in den Kernspins. Allerdings verschwindet die Rückkopplung des Elektrons auf die Kerspins im thermodynamischen Grenzfall aufgrund der asymmetrischen Kopplung eines einzelnen Elektronspins und einer grossen Kernspinumgebung. Unter der Annahme einer konstant bleibenden Kopplung beobachten wir dennoch, dass die spektrale Lücke am kritischen Punkt im thermodynamischen Grenzfall verschwindet und damit auf eine nicht-analytische Veränderung des stationären Zustandes am kritischen Punkt hinweist. iv
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