We suggest that nonequilibrium Bose-Einstein condensates may occur in time-periodically driven interacting Bose gases. Employing the model of a periodically forced bosonic Josephson junction, we demonstrate that resonance-induced ground-state-like many-particle Floquet states possess an almost perfect degree of coherence, as corresponding to a mesoscopically occupied, explicitly time-dependent single-particle orbital. In marked contrast to the customary time-independent Bose-Einstein condensates, the order parameter of such systems is destroyed by violent fluctuations when the particle number becomes too large, signaling the nonexistence of a proper mean-field limit.
We suggest that nonequilibrium Bose-Einstein condensates may occur in time-periodically driven interacting Bose gases. Employing the model of a periodically forced bosonic Josephson junction, we demonstrate that resonance-induced ground-state-like many-particle Floquet states possess an almost perfect degree of coherence, as corresponding to a mesoscopically occupied, explicitly time-dependent single-particle orbital. In marked contrast to the customary time-independent Bose-Einstein condensates, the order parameter of such systems is destroyed by violent fluctuations when the particle number becomes too large, signaling the nonexistence of a proper mean-field limit.
We argue that a Bose–Einstein condensate can be transformed into a Floquet condensate, that is, into a periodically time-dependent many-particle state possessing the coherence properties of a mesoscopically occupied single-particle Floquet state. Our reasoning is based on the observation that the denseness of the many-body system’s quasienergy spectrum does not necessarily obstruct effectively adiabatic transport. Employing the idealized model of a driven bosonic Josephson junction, we demonstrate that only a small amount of Floquet entropy is generated when a driving force with judiciously chosen frequency and maximum amplitude is turned on smoothly.
A solvable model of a periodically driven trapped mixture of Bose-Einstein condensates, consisting of N1 interacting bosons of mass m1 driven by a force of amplitude fL,1 and N2 interacting bosons of mass m2 driven by a force of amplitude fL,2, is presented. The model generalizes the harmonic-interaction model for mixtures to the time-dependent domain. The resulting many-particle ground Floquet wavefunction and quasienergy, as well as the time-dependent densities and reduced density matrices, are prescribed explicitly and analyzed at the many-body and mean-field levels of theory for finite systems and at the limit of an infinite number of particles. We prove that the time-dependent densities per particle are given at the limit of an infinite number of particles by their respective mean-field quantities, and that the time-dependent reduced one-particle and two-particle density matrices per particle of the driven mixture are 100% condensed. Interestingly, the quasienergy per particle does not coincide with the mean-field value at this limit, unless the relative center-of-mass coordinate of the two Bose-Einstein condensates is not activated by the driving forces fL,1 and fL,2. As an application, we investigate the imprinting of angular momentum and its fluctuations when steering a Bose-Einstein condensate by an interacting bosonic impurity and the resulting modes of rotations. Whereas the expectation values per particle of the angular-momentum operator for the many-body and mean-field solutions coincide at the limit of an infinite number of particles, the respective fluctuations can differ substantially. The results are analyzed in terms of the transformation properties of the angular-momentum operator under translations and boosts, and as a function of the interactions between the particles. Implications are briefly discussed.
We demonstrate longtime coherent subharmonic motion of a many-boson system subjected to an external time-periodic driving force. The underlying mechanism is exemplified numerically through analysis of a periodically driven Bose-Hubbard dimer, and clarified conceptually by semiclassical requantization of invariant tubes pertaining to the system’s mean-field description. In this way, one arrives at pre-Floquet states that relate to the actual many-body Floquet states in a manner similar to the relation of site-localized Wannier states to lattice-extended Bloch states in solid-state physics. It is argued that even high-order subharmonic response can be systematically engineered, and be observed experimentally, with weakly interacting Floquet condensates comprising a sufficiently large number of particles.
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.
Experiments on the nonequilibrium dynamics of an isolated Bose-Einstein condensate (BEC) in a magnetic double-well trap exhibit a puzzling divergence: While some show dissipation-free Josephson oscillations, others find strong damping. Such damping in isolated BECs cannot be understood on the level of the coherent Gross-Pitaevskii dynamics. Using the Keldysh functional-integral formalism, we describe the time-dependent system dynamics by means of a multimode BEC coupled to fluctuations (single-particle excitations) beyond the Gross-Pitaevskii saddle point. We find that the Josephson oscillations excite an excess of fluctuations when the effective Josephson frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{J}$ is in resonance with the effective fluctuation energy ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}}_{m}$, where both ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{J}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}}_{m}$ are strongly renormalized with respect to their noninteracting values. By evaluating and using the model parameters for the respective experiments, we describe quantitatively the presence or absence of damping.
We show that the orbital Josephson effect appears in a wide range of driven atomic Bose-Einstein condensed systems, including quantum ratchets, double wells and box potentials. We use three separate numerical methods: the Gross-Pitaevskii equation, exact diagonalization of the few-mode problem and the multi-configurational time-dependent Hartree for bosons algorithm. We establish the limits of mean-field and few-mode descriptions, demonstrating that the few-mode approximation represents the full many-body dynamics to high accuracy in the weak driving limit. Among other quantum measures, we compute the instantaneous particle current and the occupation of natural orbitals. We explore four separate dynamical regimes, the Rabi limit, chaos, the critical point and self-trapping; a favorable comparison is found even in the regimes of dynamical instabilities or macroscopic quantum self-trapping. Finally, we present an extension of the (t, t')-formalism to general time-periodic equations of motion, which permits a systematic description of the long-time dynamics of resonantly driven many-body systems, including those relevant to the orbital Josephson effect.