Maximum-Entropy Method Applied to Micro- and Nanolasers Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) genehmigt durch die Fakultät für Naturwissenschaften der Otto-von-Guericke-Universität Magdeburg von M. Sc. Boris Melcher geb. am 04. August 1990 in Kara-Balta Gutachter: Prof. Dr. Jan Wiersig Jun.-Prof. Dr. Doris Reiter eingereicht am: 15. Dezember 2020 verteidigt am: 28. April 2021 Abstract During the last decades, the principle of maximum entropy established itself as a fundamen- tal principle of consistent reasoning and hence has been successfully applied in many fields inside and outside of physics. Whenever only partial information is available it is an ideal tool to derive full statistical distributions or, in the case of quantum-mechanical descriptions, the full density matrix in a least-biased manner by exactly reproducing the given a-priori information while being maximally non-committal otherwise. In this thesis the maximum-entropy method is applied to the context of micro- and nanolasers. While tremendous improvements concerning the miniaturization and efficiency of these lasers have been achieved in the recent years, their light is still most often characterized in terms of the first statistical moments of the photon-number distribution, e.g., the light intensity and auto-correlation function. Lately, however, direct measurement of the full photon distribution became available experimentally, and theoretically its availability is highly desirable since it contains the full information about the system and therefore enables a deeper insight into the underlying physical processes. The application of the maximum-entropy method in this work is threefold. Firstly, a birth- death model that describes the interaction of a cavity light-field with a quantum-dot system is considered as a benchmark. The full photon-number distribution which is available via conventional methods here is compared to the maximum-entropy distributions that are obtained when known photon moment values are used as input information. Secondly, a combination of the maximum-entropy method and equation-of-motion approaches is proposed where output quantities of the latter are used as input for the former. Apart from the access to the full photon-number distribution, the method also provides the possibility to define an unambiguous threshold even for cases where usual definitions fail. Lastly, the maximum-entropy method is used as a stand-alone approach without the need for moment values from external sources. Instead, knowledge of the stationarity of several distinct observables is used as an input to derive the least biased steady state. By doing so, the many-particle hierarchy problem that arises in conventional equation-of-motion techniques is circumvented. Moreover, the numerical determination of the full density matrix is enabled and access to all relevant expectation values and the full statistics is obtained. The results are compared to the numerically exact solution of the von Neumann-Lindblad equation for a four-level system resonantly coupled to a single cavity mode. Moreover, usage of the maximum-entropy method as a trial-and-error approach to identify the relevant processes of quantum systems is exemplified. iii Contents 1 Introduction 1 2 Maximum-Entropy Method 5 2.1 Historical Overview . 5 2.2 Maximum-Entropy Method with Moment Constraints . 14 2.2.1 Shannon Entropy . 15 2.2.2 The Maximum-Entropy Distribution . 18 2.2.3 Bounds on the Maximum-Entropy Distribution . 20 2.2.4 Implementation and Numerical Remarks . 23 2.3 Maximum-Entropy Method for Quantum Systems . 26 2.3.1 Von Neumann Entropy . 26 2.3.2 The Maximum-Entropy Density Matrix . 28 2.3.3 Many-Particle Hierarchy Problem and Least Biased Steady State . 30 2.3.4 Implementation and Numerical Remarks . 33 3 Maximum-Entropy Method with Moment Constraints 37 3.1 Benchmark Model . 37 3.1.1 Birth-Death Model . 38 3.1.2 Photon and Emitter Statistics . 40 3.1.3 Existence of the Distribution . 43 3.1.4 Threshold Definition Comparison . 46 3.1.5 Characterization of the Emitted Light by Entropy . 49 3.2 Superthermal Photon Bunching . 52 3.2.1 Superthermal Photon Bunching for Single-Mode Distributions . 53 3.2.2 Anti-Correlated Two-Mode Distributions . 56 3.3 Chapter Conclusion . 58 4 Combining Equation-of-Motion Techniques and the Maximum-Entropy Method 61 4.1 Semiconductor Model . 61 4.2 Nanolasers with Extended Gain Media . 66 4.3 Chapter Conclusion . 73 v 5 Maximum-Entropy Method for Quantum Systems 75 5.1 Benchmark Model . 76 5.1.1 Four-Level Quantum-Dot Microcavity Laser . 76 5.1.2 Input Information and Observation Levels . 78 5.1.3 Numerical Results . 81 5.2 Outlook . 87 5.2.1 Bimodal Quantum-Dot Microcavity Laser . 87 5.2.2 Excitation with a Second Light Field . 91 5.2.3 General Outlook . 92 5.3 Chapter Conclusion . 94 6 Conclusion 95 A Appendix 97 A.1 Upper Bound of the Second-Order Maximum-Entropy Distribution . 97 A.2 Choice of Selfadjoint Operators as Input Information . 98 A.3 Choice of the Maximum Photon Number for Numerical Calculations . 100 A.4 Entropy of a Thermal Distribution . 102 Bibliography 103 vi Introduction 1 In the middle of the 19th century, a time when heat machines were already widely used, 28-year-old Sadi Carnot not only laid the foundation for the physical sub-field of thermo- dynamics but also was the first one to encounter the concept of entropy (Carnot, 1824; Volkenstein, 2009; Carnot, 2012). Afterwards, the attempt to describe the dynamics of heat phenomena quickly expanded to also include entropy’s statistical nature as well as its role as an uncertainty measure. Many well-known scientists such as Clausius, Boltzmann, Maxwell and Gibbs in the earlier years and Shannon, von Neumann and Jaynes in the later years were involved in this process that finally also lead to the principle of maximum entropy (Clausius, 1850a; Clausius, 1850b; Clausius, 1865; Gibbs, 1902; Shannon, 1948; Bril- louin, 1951; Jaynes, 1957a; Jaynes, 1957b; Jaynes, 1965; Ehrenberg, 1967; Garber, 1970; Tribus & McIrvine, 1971; Von Neumann, 1996; Lieb & Yngvason, 1999; Jaynes, 2003; Downarowicz, 2007; Griffin, 2008; Volkenstein, 2009; Pressé et al., 2013; Uffink, 2017; Chehade & Vershynina, 2019). With the more recent discoveries, the procedure of entropy maximization is no longer merely seen as maximizing uncertainty or minimizing bias but rather as a way to draw consistent inferences from partial data (Griffin, 2008; Volkenstein, 2009; Pressé et al., 2013; Uffink, 2017; Jizba & Korbel, 2019; Jizba & Korbel, 2020). With that, the maximum-entropy principle exceeded the realm of physics and established itself as a form of “extended logic”1. Because the maximum-entropy method is an ideal tool to handle incomplete information and deduce self-consistent conclusions without artificially including any bias, it has since then been used in various fields inside and outside of physics, such as astronomy, biology, economics, finance, and quantum chemistry among others (Pressé et al., 2013; Zhou et al., 2013; He & Kolovos, 2018; Jizba & Korbel, 2019; Tsallis, 2019; Chanda et al., 2020). In the same time that was required to arrive at today’s notion of the principle of maximum entropy, an enormous information technological transformation took place. While in 1824, the year of Carnot’s publication, great effort was put into the development of the first telegraph (Meadow, 2002), we are now living in the digital Zettabyte Era where the annual IP traffic exceeds at least one Zettabyte (1 ZB = 109 TB = 1021 B) (Cisco, 2019; Ma & Oulton, 2019). Handling this huge amount of digital information accordingly consumes a substantial percentage of the worldwide electricity (Malmodin & Lundén, 2018; Ma & Oulton, 2019) and has a large impact on the environment with an estimated 2.3 % of 1Jaynes, 2003, p. xxii. 1 the global green house gas emission stemming from the information and communication technologies sector in 2020 (Global e-Sustainability Initiative, 2012). Fortunately, the ongoing effort to decrease the emission already resulted in a more optimistic trend with an estimation of a percentage of roughly 2.0 % for 2030 (Global e-Sustainability Initiative, 2015). Nevertheless, the search for energy-efficient technologies is still of major interest both ecologically and economically. One of its many aspects is the ongoing miniaturization of lasing devices that are, for instance, used for data transmission between individual chips or different sections within chips (Ma & Oulton, 2019). Concerning laser systems a plethora of different aspects, e.g., the light-matter coupling strength (Vahala, 2003; He et al., 2013; Dovzhenko et al., 2018; Frisk Kockum et al., 2019), the quality factor of the cavity mode (Vahala, 2003; Reitzenstein & Forchel, 2010; He et al., 2013), and the power efficiency (Ma & Oulton, 2019) play a role and are currently exploited to improve the desired properties of those systems. As gain medium quantum-dots are among the structures considered most often (Noda, 2006; Norman et al., 2019). In those nanometer-sized objects the charge carriers are locally confined in all three spatial dimensions which can lead to strong light-matter interaction and high gain (Noda, 2006; Reitzenstein, 2012; Norman et al., 2019). Apart from that, the confining potential and the discrete energy levels of these artificial atoms can be tailored by, e.g., different choices of material, the quantum dot size, shape or by application of strain (Reitzenstein, 2012; Huo et al., 2013; Huo et al., 2014a; Huo et al., 2014b; Goldmann, 2014; Schumann, 2016; Melcher, 2016; Norman et al., 2019). The advances in the area of laser cavities, on the other hand, where improved manufacturing techniques allowed for increasingly higher quality factors and smaller mode volumes, further facilitated the possibility to build lasers where the better part of the spontaneous emission is directed into the laser mode (Vahala, 2003; Noda, 2006; Strauf et al., 2006; Thyrrestrup et al., 2010; He et al., 2013; Ma & Oulton, 2019).