Bose-Einstein condensation, the macroscopic occupation of a single quantum state, appears in equilibrium quantum statistical mechanics and persists also in the hydrodynamic regime close to equilibrium. Here we show that even when a degenerate Bose gas is driven into a steady state far from equilibrium, where the notion of a single-particle ground state becomes meaningless, Bose-Einstein condensation survives in a generalized form: the unambiguous selection of an odd number of states acquiring large occupations. Within mean-field theory we derive a criterion for when a single state and when multiple states are Bose selected in a noninteracting gas. We study the effect in several driven-dissipative model systems, and propose a quantum switch for heat conductivity based on shifting between one and three selected states.
Bose-Einstein condensation, the macroscopic occupation of a single quantum state, appears in equilibrium quantum statistical mechanics and persists also in the hydrodynamic regime close to equilibrium. Here we show that even when a degenerate Bose gas is driven into a steady state far from equilibrium, where the notion of a single-particle ground state becomes meaningless, Bose-Einstein condensation survives in a generalized form: the unambiguous selection of an odd number of states acquiring large occupations. Within mean-field theory we derive a criterion for when a single state and when multiple states are Bose selected in a noninteracting gas. We study the effect in several driven-dissipative model systems, and propose a quantum switch for heat conductivity based on shifting between one and three selected states.
Bose-Einstein condensation, the macroscopic occupation of a single quantum state, lies at the heart of fascinating quantum phenomena like superfluidity in liquid Helium, superconductivity in solids, or matter-wave interference in ultracold quantum gases. It is considered a consequence of equilibrium quantum statistical mechanics, persisting also in the hydrodynamic regime close to equilibrium. Here we show that even when a degenerate Bose gas is driven into a steady state far from equilibrium, where the notion of a single-particle ground state becomes meaningless, Bose-Einstein condensation persists in a generalized form: the unambiguous selection of an odd number of states acquiring large occupations. We derive a criterion for when a single and when multiple states are Bose selected in a non-interacting gas. We study the effect in several driven-dissipative model systems, and propose a quantum switch for heat conductivity based on shifting between one and three selected states.
We investigate nonequilibrium steady states of driven-dissipative ideal quantum gases of both bosons and fermions. We focus on systems of sharp particle number that are driven out of equilibrium either by the coupling to several heat baths of different temperature or by time-periodic driving in combination with the coupling to a heat bath. Within the framework of (Floquet-)Born-Markov theory, several analytical and numerical methods are described in detail. This includes a mean-field theory in terms of occupation numbers, an augmented mean-field theory taking into account also nontrivial two-particle correlations, and quantum-jump-type Monte Carlo simulations. For the case of the ideal Fermi gas, these methods are applied to simple lattice models and the possibility of achieving exotic states via bath engineering is pointed out. The largest part of this work is devoted to bosonic quantum gases and the phenomenon of Bose selection, a nonequilibrium generalization of Bose condensation, where multiple single-particle states are selected to acquire a large occupation [Phys. Rev. Lett. 111, 240405 (2013)]. In this context, among others, we provide a theory for transitions where the set of selected states changes, describe an efficient algorithm for finding the set of selected states, investigate beyond-mean-field effects, and identify the dominant mechanisms for heat transport in the Bose-selected state.
Bose-Einstein condensation is a collective quantum phenomenon where a macroscopic number of bosons occupies the lowest quantum state.For fixed temperature, bosons condense above a critical particle density.This phenomenon is a consequence of the Bose-Einstein distribution which dictates that excited states can host only a finite number of particles so that all remaining particles must form a condensate in the ground state.This reasoning applies to thermal equilibrium.We investigate the fate of Bose condensation in nonisolated systems of noninteracting Bose gases driven far away from equilibrium.An example of such a driven-dissipative scenario is a Floquet system coupled to a heat bath.In these time-periodically driven systems, the particles are distributed among the Floquet states, which are the solutions of the Schrödinger equation that are time periodic up to a phase factor.The absence of the definition of a ground state in Floquet systems raises the question, whether Bose condensation survives far from equilibrium.We show that Bose condensation generalizes to an unambiguous selection of multiple states each acquiring a large occupation proportional to the total particle number.In contrast, the occupation numbers of nonselected states are bounded from above.We observe this phenomenon not only in various Floquet systems, i.a.time-periodically-driven quartic oscillators and tight-binding chains, but also in systems coupled to two baths where the population of one bath is inverted.In many cases, the occupation numbers of the selected states are macroscopic such that a fragmented condensation is formed according to the Penrose-Onsager criterion.We propose to control the heat conductivity through a chain by switching between a single and several selected states.Furthermore, the number of selected states is always odd except for fine-tuning.We provide a criterion, whether a single state (e.g., Bose condensation) or several states are selected.In open systems, which exchange also particles with their environment, the nonequilibrium steady state is determined by the interplay between the particle-number-conserving intermode kinetics and particle-number-changing pumping and loss processes.For a large class of model systems, we find the following generic sequence when increasing the pumping: For small pumping, no state is selected.The first threshold, where the stimulated emission from the gain medium exceeds the loss in a state, is equivalent to the classical lasing threshold.Due to the competition between gain, loss and intermode kinetics, further transitions may occur.At each transition, a single state becomes either selected or deselected.Counterintuitively, at sufficiently strong pumping, the set of selected states is independent of the details of the gain and loss.Instead, it is solely determined by the intermode kinetics like in closed systems.This implies equilibrium condensation when the intermode kinetics is caused by a thermal environment.These findings agree well with observations of exciton-polariton gases in microcavities.In a collaboration with experimentalists, we observe and explain the pump-powerdriven mode switching in a bimodal quantum-dot micropillar cavity.v Zusammenfassung Die Bose-Einstein-Kondensation ist ein Quantenphänomen, bei dem eine makroskopische Zahl von Bosonen den tiefsten Quantenzustand besetzt.Die Teilchen kondensieren, wenn bei konstanter Temperatur die Teilchendichte einen kritischen Wert übersteigt.Da die Besetzungen von angeregten Zuständen nach der Bose-Einstein-Statistik begrenzt sind, bilden alle verbleibenden Teilchen ein Kondensat im Grundzustand.Diese Argumentation ist im thermischen Gleichgewicht gültig.In dieser Arbeit untersuchen wir, ob die Bose-Einstein-Kondensation in nicht wechselwirkenden Gasen fern des Gleichgewichtes überlebt.Diese Frage stellt sich beispielsweise in Floquet-Systemen, welche Energie mit einer thermischen Umgebung austauschen.In diesen zeitperiodisch getriebenen Systemen verteilen sich die Teilchen auf Floquet-Zustände, die bis auf einen Phasenfaktor zeitperiodischen Lösungen der Schrödinger-Gleichung.Die fehlende Definition eines Grundzustandes wirft die Frage nach der Existenz eines Bose-Kondensates auf.Wir finden eine Generalisierung der Bose-Kondensation in Form einer Selektion mehrerer Zustände.Die Besetzung in jedem selektierten Zustand ist proportional zur Gesamtteilchenzahl, während die Besetzung aller übrigen Zustände begrenzt bleibt.Wir beobachten diesen Effekt nicht nur in Floquet-Systemen, z.B. getriebenen quartischen Fallen, sondern auch in Systemen die an zwei Wärmebäder gekoppelt sind, wobei die Besetzung des einen invertiert ist.In vielen Fällen ist die Teilchenzahl in den selektierten Zuständen makroskopisch, sodass nach dem Penrose-Onsager Kriterium ein fragmentiertes Kondensat vorliegt.Die Wärmeleitfähigkeit des Systems kann durch den Wechsel zwischen einem und mehreren selektierten Zuständen kontrolliert werden.Die Anzahl der selektierten Zustände ist stets ungerade, außer im Falle von Feintuning.Wir beschreiben ein Kriterium, welches bestimmt, ob es nur einen selektierten Zustand (z.B.Bose-Kondensation) oder viele selektierte Zustände gibt.In offenen Systemen, die auch Teilchen mit der Umgebung austauschen, ist der stationäre Nichtgleichgewichtszustand durch ein Wechselspiel zwischen der (Teilchenzahl-erhaltenden) Intermodenkinetik und den (Teilchenzahl-ändernden) Pump-und Verlustprozessen bestimmt.Für eine Vielzahl an Modellsystemen zeigen wir folgendes typisches Verhalten mit steigender Pumpleistung: Zunächst ist kein Zustand selektiert.Die erste Schwelle tritt auf, wenn der Gewinn den Verlust in einer Mode ausgleicht und entspricht der klassischen Laserschwelle.Bei stärkerem Pumpen treten weitere Übergänge auf, an denen je ein einzelner Zustand entweder selektiert oder deselektiert wird.Schließlich ist die Selektion überraschenderweise unabhängig von der Charakteristik des Pumpens und der Verlustprozesse.Die Selektion ist vielmehr ausschließlich durch die Intermodenkinetik bestimmt und entspricht damit den oben beschriebenen geschlossenen Systemen.Ist die Kinetik durch ein thermisches Bad hervorgerufen, tritt wie im Gleichgewicht eine Grundzustands-Kondensation auf.Unsere Theorie ist in Übereinstimmung mit experimentellen Beobachtungen von Exziton-Polariton-Gasen in Mikrokavitäten.In einer Kooperation mit experimentellen Gruppen konnten wir den Modenwechsel in einem bimodalen Quantenpunkt-Mikrolaser erklären.
Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.
We derive a theory for Bose condensation in nonequilibrium steady states of bosonic quantum gases that are coupled both to a thermal heat bath and to a pumped reservoir (or gain medium), while suffering from loss. Such a scenario describes photonic many-body systems such as exciton-polariton gases. Our analysis is based on a set of kinetic equations for a gas of noninteracting bosons. By identifying a dimensionless scaling parameter controlling the boson density, we derive a sharp criterion for which system states become selected to host a macroscopic occupation. We show that with increasing pump power, the system generically undergoes a sequence of nonequilibrium phase transitions. At each transition, a state either becomes or ceases to be Bose selected (i.e., to host a condensate): The state which first acquires a condensate when the pumping exceeds a threshold is the one with the largest ratio of pumping to loss. This intuitive behavior resembles simple lasing. In the limit of strong pumping, the coupling to the heat bath becomes dominant so that eventually the ground state is selected, corresponding to equilibrium(like) Bose condensation. For intermediate pumping strengths, several states become selected, giving rise to fragmented nonequilibrium Bose condensation. We compare these predictions to experimental results obtained for exciton polaritons in a double-pillar structure [M. Galbiati et al., Phys. Rev. Lett. 108, 126403 (2012)] and find good agreement. Our theory, moreover, predicts that the reservoir occupation is clamped at a constant value whenever the system hosts an odd number of Bose condensates.
We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number N and total momentum P of all particles are fixed. Using the saddle-point method, we derive the large-N analytical approximations for partition function, free energy, and statistical distributions of occupation numbers of different single-particle energy levels. At temperatures below the critical point of phase transition, we predict, in some ranges of P, fragmentation of the condensate, when more than one single-particle level is macroscopically occupied. The occupation number distributions have approximately Gaussian shapes for the levels hosting the condensate, and exponential shapes for other, noncondensate levels. Our analysis demonstrates breaking of Galilean invariance of moving finite-temperature many-particle system in the presence of Bose-Einstein condensation and extends the theory of moving and rotating quantum systems to the finite-temperature large-N limit.