Quantum-Classical Transition in Many-Body Systems: Indistinguishability, Interference and Interactions

Recently, we discovered an exact trace duality for kicked, one dimensional spin chains. It expresses traces of the unitary evolution operator for N spins and T time-steps by traces of an operator whose dimension is determined by T instead of N. This simplification allows us to address large spin systems at practically arbitrary N. With this at hand, we can perform a semi-classical analysis of a many-body system, determining (action-)spectra and their correspondence to classical periodic orbits. In this initial study we illustrate challenges of mixed classical dynamics, the exponential orbit count proliferation with N, and possible pathways to overcome these obstacles.

We demonstrate that in a standard thermo-electric nanodevice the current and heat flows are not only dictated by the temperature and potential gradient but also by the external action of a local quantum observer that controls the coherence of the device. Depending on how and where the observation takes place the direction of heat and particle currents can be independently controlled. In fact, we show that the current and heat flow can go against the natural temperature and voltage gradients. Dynamical quantum measurement offers new possibilities for the control of quantum transport far beyond classical thermal reservoirs. Through the concept of local projections, we illustrate how we can create and directionality control the injection of currents (electronic and heat) in nanodevices. This scheme provides novel strategies to construct quantum devices with application in thermoelectrics, spintronic injection, phononics, and sensing among others. In particular, highly efficient and selective spin injection might be achieved by local spin projection techniques.

Phase transitions to absorbing states are among the simplest examples of critical phenomena out of equilibrium. The characteristic feature of these models is the presence of a fluctuationless configuration which the dynamics cannot leave, which has proved a rather stringent requirement in experiments. Recently, a proposal to seek such transitions in highly tuneable systems of cold atomic gases offers to probe this physics and, at the same time, to investigate the robustness of these transitions to quantum coherent effects. Here we specifically focus on the interplay between classical and quantum fluctuations in a simple driven open quantum model which, in the classical limit, reproduces a contact process, which is known to undergo a continuous transition in the "directed percolation" universality class. We derive an effective long-wavelength field theory for the present class of open spin systems and show that, due to quantum fluctuations, the nature of the transition changes from second to first order, passing through a bicritical point which appears to belong instead to the "tricritical directed percolation" class.

We study exciton condensate formation and coherent transport in a sys- tem of two graphene bilayers that are separated by hexagonal boron-nitride. A variety of excitonic-phases has been identified by varying separately the carrier density in the graphene sheets of each bilayer as well as the distance between the bilayer sheets [1]. The richness of the phase diagram could be the key to understand the interesting transport properties observed in recent experiments [2,3]. We aim to explain some key experimental features in terms of peculiar transport properties associated to excitonic condensates [4]. [1] J. Su and A.H. MacDonald, arXiv:1611.06410 (2016). [2] J.I.A. Li et al., Phys. Rev. Lett., 117 046802 (2016). [3] K. Lee et al., Phys. Rev. Lett. 117, 046803 (2016). [4] M. Rontani & L. J. Sham, Phys. Rev. Lett. 94, 186404 (2005).

Feynman [R. P. Feynman, Lectures on Physics (Addison-Wesley, Reading, MA, 1963), Vol. 1.] reignited the interest in directed transport, i.e. unbiased transport phenomena in systems which are driven out of equilibrium. There is a wealth of fields for their application such as in biology, nanotechnology, chemistry, and Bose-Einstein condensates for example. Among the very many alternatives, we focus in ratchets with dissipation which are generally associated with a classical asymmetric chaotic attractor; the quantum versions lead to interesting applications in cold atoms. The classical and quantum aspects of the parameter space of these systems have been the subject of very recent and interesting developments [Phys. Rev. E 93, 042133 (2016), Phys. Rev. E 91, 052908 (2015)]. We will give a glimpse on them, explaining the role of the isoperiodic stable structures (ISSs, Lyapunov stable islands), fundamental for the current properties. The quantum counterparts of these structures (QISSs) have proven to be very well approximated by means of a thermal coarse graining of the classical dynamical equations. The details on this correspondence mechanism open new lines of research and possibly provide with interesting answers to some quantum to classical correspondence questions [arXiv:1604.02743].

We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain (t−V-model). Our observable is the time-dependent density propagator, Π_ε(x,t), calculated in windows of different energy density, ε, of the many-body Hamiltonian and at different disorder strengths, W, not exceeding the critical value W_c. The width Δx_ε(t) of Π_ε(x,t) exhibits a behavior dlnΔx_ε(t)/dlnt=β_ε(t), where the exponent function β_ε(t)≲1/2 is seen to depend strongly on L at all investigated parameter combinations. (i) We cannot confirm the existence of a region in phase space that exhibits (genuine) subdiffusive dynamics in the sense that β_ε<1/2 is numerically fixed in the limit of large L. Instead, subdiffusion might possibly be transient, only, finally giving way to conventional diffusive behavior with β_ε=1/2. (ii) Similarly, we cannot confirm the existence of many-body mobility edges deep in the delocalized phase. (iii) (Transient) subdiffusion 0<β_ε(t)≲1/2, coexists with an enhanced probability for returning to the origin, Π_ε(0,t), decaying much slower than 1/Δx_ε(t). Correspondingly, the spatial decay of Πε(x,t) is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.

Authors: Gabriel Dufour, Tobias Brünner, Maximilian Dirkmann, Alberto Rodriguez, Andreas Buchleitner The dynamics of ensembles of quantum particles depends on whether these particles can be distinguished from one another. Indeed, the requirement that many-body states be symmetric under the exchange of identical particles leads to intricate interference effects. These effects have mainly been investigated in the case of non-interacting particles, such as photons in interferometers, but they also play a role in the behaviour of interacting particles, e.g. atoms trapped in optical lattices. We investigate the ways in which the dynamics of distinguishable and indistinguishable particles differ. To do so, we consider ensembles of bosons condensed into a finite number of modes and which can belong to one of several mutually distinguishable ``species''. We study the structure of the corresponding Hilbert space and its consequences for the dynamics. Moreover, we identify a suitable measure of the level of distinguishability in the system. These results are illustrated by a study of the dynamics of a two-component Bose-Einstein condensate in a double-well.

In order to investigate general properties of interacting bosonic gases we present a formalism to calculate thermodynamic properties and the density of states by means of short-time propagation and compare our analytical predictions against quantum integrable models using Bethe ansatz techniques. As an essential input of our approach, we were able to construct the many-body propagator for a one-dimensional free bosonic gas with delta interactions of variable strength. Using this propagator we can give analytical expressions for the smooth part of the many-body density of states as well as spatial two point correlations for the Lieb-Liniger model. We present explicit analytical results for the hight temperature regime that are nonperturbative in the interaction strength, while corrections for lower temperatures can be found systematically. A perturbative approach for arbitrary temperatures using Matsubara theory is presented, which is valid in the weakly interacting limit.

The Majorization Principle is a fundamental statement ruling the dynamics of information processing in optimal and efficient quantum algorithms. They are conjectured to obey a quantum arrow of time governed by the Majorization Principle: the probability distribution associated to the outcomes gets ordered step-by-step until achieving the result of the computation. The Majorization Principle has been observed experimentally in two quantum algorithms, namely the quantum fast Fourier transform and a recently introduced validation protocol for the certification of genuine many-boson interference. The demonstration has been performed through integrated 3-D photonic circuits fabricated via femtosecond laser writing technique, which allows to monitor unambiguously the effects of majorization along the execution of the algorithms. Since the quantum validation protocol is based on genuine bosonic interference, it is fundamental to investigate how partial photon distinguishability influences the correspondent majorization’s relations: specifically, a quantum-to-classical transition is observed while varying the time delay between the injected photons. The measured observables provide a strong indication that the Majorization Principle holds true for this wide class of quantum algorithms, thus paving the way for a general tool to design new optimal algorithms with a quantum speedup.

Phase space crystals is a novel and mathematically rather challenging subject in physics. Phase space crystals can be created by driving suitable classical or quantum mechanical systems in a specific periodic way. In this way it is possible to create systems with a variety of effective Hamiltonians which are otherwise difficult to realize (such as quasicrystals), and they can be varied by tuning the driving parameters. Phase space crystals also provide a new way to study topological phenomena in physics. They differ from lattices in real space since their coordinate systems, i.e., the phase space, has a non-commutative geometry, which naturally produces phases like those arising from magnetic fields. In this way topological insulators, which have attracted much attention recently, can be engineered. Still another option provided by phase space engineering is a controlled transition between one-dimensional and two-dimensional crystals in an effective magnetic field. The famous Hofstadter-butterfly structure in the energy spectrum thus can be realized in one-dimensional systems. The primary physical system to create phase space crystals are ultracold atoms in optical lattices.

We introduce a stability criterion for quantum statistical ensembles describing macroscopic systems. An ensemble is called "stable" when a small number of local measurements cannot significantly modify the probability distribution of the total energy of the system. We apply this criterion to lattices of spins-1/2, thereby showing that the canonical ensemble is nearly stable, whereas statistical ensembles with much broader energy distributions are not stable. In the context of the foundations of quantum statistical physics, this result justifies the use of statistical ensembles with narrow energy distributions such as canonical or microcanonical ensembles.

Understanding how isolated quantum systems thermalize has recently gathered renewed interest among theorists, thanks to the experimental realizations of such systems. As a sufficient condition for the approach to thermal equilibrium, the eigenstate thermalization hypothesis (ETH) is particularly investigated, which focuses on matrix elements of observables in the thermodynamic limit. Recent numerical studies suggest that the ETH and its finite-size corrections in nonintegrable systems coincide with the prediction of the random matrix theory (RMT). This coincidence, which we call a quantum chaos conjecture (QCC), is expected to be the underlying mechanism for the ETH. However, it is not completely understood to what extent the QCC is correct — especially how it depends on observables. In this poster, by investigating ETH corrections, we show that the QCC is true for a wide class of observables with various symmetries, including many-body operators. First, using the RMT, we predict a general form of ETH corrections, concerning the symmetries of systems and observables. We especially note that the ratios of variances between diagonal and off-diagonal matrix elements are universal and dependent only on the symmetries. Then, we numerically show that the ratios calculated in nonintegrable systems coincide with the prediction of the RMT, indicating the QCC, both for typical few-body and many-body observables. We finally remark, though, that a counterexample of the QCC exists even among simple observables, when it is related to the Hamiltonian.

Boson sampling has emerged as a tool to demonstrate the difference between quantum and classical computers and has piqued the interest of experimentalists and theoreticians. Its main advantage is that it does not require full universal control over the quantum system, which favours current photonic experimental platforms for implementation. However, they suffer from a scaling problem because of probabilistic state generation. We circumvent this problem by devising a boson sampling protocol using Gaussian states and show that this a more general type of photonic boson sampler. First we derive a new result that relates the probability of a photon pattern generated by a general Gaussian state and show that its depend upon a matrix function called the hafnian. This leads to a feasible experimental method for a photonic boson sampler with significant numbers of photons, going clearly far beyond the limitations of previous scenarios.

We consider a one-dimensional model of Bosons with attractive interactions which is known to display a quantum phase transition (QPT) at a critical value of the interaction strength $\alpha$. A semiclassical quantization, where a large number of particles $N$ plays the role of small $\hbar$, allows us to find the parameters of the QPT in terms of quantized orbits passing through a separatrix in classical phase space. Moreover it allows us to analytically quantify not only the scaling of the ground state gap with $N$ right at the critical $\alpha$ (explaining the numerical findings in [1]), but also the high-density spectrum around the excited state QPT for larger couplings. We extract the exact time-scale of the latter - also known as the scrambling time - and rigurously show that it is related to a stability exponent of the underlying classical dynamics. Finally, the universal applicability of our ideas whenever the QPT is related to similar classical behaviour, leads us to expect that this semiclassical picture represents a broad class of QPT with effective one-dimensional descriptions (e.g. Dicke model[2], Lipkin-Meshkov-Glick model[3]). [1] R Kanamoto and H Saito and M Ueda, Phys. Rev. A 67, 013608 (2003) [2] T Brandes, Phys. Rev. E 88, 032133 (2013) [3] P Ribeiro and J Vidal and R Mosseri, Phys. Rev. E 78, 021106 (2008)

We experimentally demonstrate linear optical computation of the permanent of a Hermitian positive semi-definite matrix. By measuring the second-order correlation of pulsed thermal light, we show that it is possible to compute the permanent of a 2 x 2 Hermitian positive semi-definite matrix defined by a beam splitter. We also show the results of our on-going experiment on linear optical computation of the permanent of a 4 x 4 matrix. Our approach can be scaled up for an m x m matrix by utilizing integrated photonics circuits.

We experimentally investigate quantum polarization of three-photon states. A number of representative three-photon states, including, pure states, mixed states, and three-photon N00N states, are prepared and tested for their quantum polarization properties, i.e., rotational invariance of mean, variance, and skewness of the quantized Stokes operators. Our results reveal interesting properties of quantum polarization, e.g., maximum-uncertainty and hidden polarization.

We analyze theoretically the ground state of weakly interacting bosons in the flux ladder -- the system that has been recently realized by means of ultacold atoms in the specially designed optical lattice [M. Atala, et al., Nat. Phys. 10, 588 (2014)]. It is shown that, for the system parameters corresponding to the so-called `vortex phase', the ground state is a fragmented condensate. We study the Bogoliubov depletion of this condensate and discuss the role of boundary conditions.

We present our novel approach to the ensemble averaged dynamics of disordered quantum systems using the framework of quantum master equations. The latter allows for a full characterization of the coherent and incoherent contents of the ensemble averaged dynamics on transient and asymptotic time scales. In the cases of spectral disorder (induced e.g. by intensity fluctuations of experimental control fields) and of isotropically disordered eigenvector distributions (i.e. random unitary ensembles), we have derived exact master equations, which are valid for all times. Furthermore, in the limit of short-times, an explicit form valid for any disorder distribution was derived. Finally, we have recently used standard perturbation theory to capture the combined effect of spectral disorder and a perturbative potential. Therewith, we derived perturbative disorder master equations for systems such as finite-size tilted Anderson-like models, tilted Bose-Hubbard Hamiltonians, frozen Rydberg gases or biology-inspired networks. Moreover, we can show that our approximations schemes can be translated into the setting of open quantum systems: the short-time limit is related to the Born-Markov approximation and the perturbative approach is a TCL-technique adapted for disordered systems.

We investigate the switching of the steady-state lasing mode, occurring in bimodal lasers when varying the pump power. Comparing experimental data to theoretical results, we identify the underlying mechanism to be based on the competition between the effective gain on the one hand and the inter-mode coupling on the other. Using simple analytical arguments, we explain that, while the largest effective gain determines the laser mode for weak pumping (just above the lasing threshold), it is the inter-mode coupling that selects the laser mode in the limit of strong pumping. We, moreover, show that the switching from one laser mode to the other occurs via an intermediate state, where both modes are lasing. Finally, we employ exact numerical simulations, to investigate the origin of two experimentally observed effects, the super-thermal intensity fluctuations of the non-lasing mode and strong anticorrelations at transitions of the laser mode.

Closed disordered interacting quantum systems can exhibit many-body localization (MBL). When disorder is sufficiently strong, such systems enter into a nonergodic regime known as the many-body localized phase, resulting in an ideal insulator with zero charge and thermal conductivities at finite energy densities. The emergent integrability of the MBL phase can be understood in terms of localized quasiparticles. As a result, the occupations of the one-particle density matrix (OPDM) in eigenstates show a Fermi-liquid like discontinuity. In this work, we numerically explore the dynamics of the MBL phase generated by a global quench from a charge density state, in terms of the the OPDM occupation spectrum. In particular, we show that in the steady state, the occupation discontinuity is smeared to a continuous distribution in a similar way as finite temperature smears the discontinuity in a Fermi liquid, but the occupation spectrum remains highly non-thermal in the thermodynamic limit.

We find quantum signatures of classical chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and hence higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by random matrix theory. We show the optimality of tomography under chaotic Hamiltonians and connect information gain in tomography, hypersensitivity to perturbation and chaos.

Originating from the context of scrambling of information in black holes[1], the question of how to diagnose chaos in the classical counterpart of a quantum system started to experience an increasing interest again. It was suggested that the averaged squared commutator $\left\langle \left[\hat{V},\hat{W}(t)\right]^\dagger \left[\hat{V},\hat{W}(t)\right] \right\rangle$, with arbitrary operators $\hat{V}$ and $\hat{W}$, is a suitable measure for chaos, in the sense that the different ordering of times are able to capture the hyperbolicity of the system, the so-called quantum butterfly effect[1]. From simple arguments involving Poisson brackets, one indeed expects this average to involve terms of exponential increase with a rate given by the classical Lyapunov exponent. This behaviour should hold up to time scales of the classical-to-quantum-crossover, a time known as Ehrenfest time or scrambling time. Many numerical studies support this claim, and also many articles focus on suggestions for the experimental realisation of the squared commutator (which is complicated due to the forward and backward propagation in time inherent in the Heisenberg picture of the operator $\hat{W}(t)$). However, analytical explanations are rare and concentrate more on the exponential behaviour and not on the Lyapunov exponent. In this presentation we want to fill this gap using semiclassical methods based on classical trajectories, which are well suited to provide a quantitative picture in chaotic systems. Using the Van-Vleck-propagator, we explicitly discuss the emergence of the Lyapunov exponent and the involved timescales for the simplified picture of the average of the commutator $\left\langle \left[\hat{V},\hat{W}(t)\right] \right\rangle$. [1] J. Maldacena, S. H. Shenker and D. Stanford, Journal of High Energy Physics, 2016:106

We study the effect of disorder on spectral properties of tubular chlorosomes which are the main light-harvesting complex in the green sulfur bacteria Cf. aurantiacus. Employing a Frenkel-exciton Hamiltonian with diagonal and off-diagonal disorder consistent with spectral and structural studies, we analyze excitonic localization and spectral statistics of the chlorosomes. A size-dependent localization-delocalization crossover is found to occur as a function of the excitonic energy. The crossover energy region coincides with the more optically active states with maximized superradiance and is, consequently, more conducive for energy transfer.

We study the generation of entanglement between two species of living bosons in a ring lattice. The entanglement is generated by managing the interaction between two distinct kind of particles. We find for a certain region of parameters that the bipartite entanglement between the two type of particles scales as a power law of the phase coherence of one of these species. The numerical findings are corroborated by exact analytical results. Our results lies at the heart of phenomena such as interference, coherence and entanglement formation.

The decoherence program explains why it is hard to observe quantum superpositions in macroscopic scales although their existence is not excluded by the laws of quantum mechanics. This explanation is based on openness of quantum systems but a question of whether and how much a macroscopic superposition as an isolated system is stable remains. Recently, there have been studies on thermalization of isolated quantum systems where expectation values of local observables after a long time give the same values with the thermal ensemble-averaged values. In this case, the eigenstate thermalization hypothesis (ETH) can be used to calculate the expectation value of an observable after thermalization. In contrast, there are many-body localized (MBL) systems which do not thermalize in spite of interactions between particles. In this work, we show that a macroscopic superposition even in an isolated system disappears when it thermalizes but it can be preserved in an MBL system. Using a well-established measure of quantum macroscopicity, we show that a macroscopic superposition of a multipartite spin system is destroyed after thermalization under the ETH with short range interactions. On the other hand, macroscopic superpositions in MBL systems can be preserved by local integrals of motion. Further, we numerically investigate a disordered Heisenberg model which alters between a thermalizing system and a many-body localizing system. Consistent results are obtained that an initial macroscopic superposition disappears if the system thermalizes but it can preserve its size as a macroscopic superposition if the system is many-body localized.

The Quantum Drift model was developed to account the decoherence induced by many-body interactions in single particle systems. It involves the introduction of a fluctuating field in a wave function dynamics.[1] By staying within a wave function framework it results useful evaluate decoherent dynamics and decoherent transport. We are going to discuss various situations for this both for integrable and non integrable systems. By choosing an appropriate basis for the action of the fields we are going to show that classical dynamics is obtained from a quantum framework. The resulting decoherence is better address in terms of the Loschmidt Echo.[2] We are going to address quantum dynamics whose classical limit is integrable and also chaotic classical dynamics. Equivalently, we may represent external environment effects and many-body pertubations in non-integrable many-body dynamics. We also are going to address this situation. In this last case, the decoherent dynamics is addressed by extending the Quantum Paralellism [3] to the inclusion in a very efficient many-body dynamics the of decoherent effect of perturbing interactions in terms of the quantum drift model. [1]Decoherent time-dependent transport beyond the Landauer-B"uttiker formulation: A quantum-drift alternative to quantum jumps, L.J. Fern\'andez-Alcázar and H.M. Pastawski, Phys. Rev. A 91, 022117 (2015) [2]Loschmidt Echo, A. Gousev, R.A. Jalabert, H.M. Pastawski and D.A. Wisniacki. Scholarpedia 7, 11687 (2012) [3]Quantum Parallelism as a Tool for Ensemble Spin Dynamics Calculations Gonzalo A. Alvarez, Ernesto P. Danieli, Patricia R. Levstein, and Horacio M. Pastawski,Phys. Rev. Lett. 101, 120503 (2008)

Neuroscience is characterised by many levels of description, from molecules over neurons and networks to brains and behaviour. Bridging the gaps between adjacent levels is often challenging. In this contribution, I want to discuss aspects of the microscopic end of this hierarchy. Ion channels are pore-like proteins located in the cell membrane of neurons and mediate transmembrane ion currents, which constitutes the core mechanism for many neuronal processes. Ion channels can be studied as large biomolecules, or in terms of effective Markov models, where only a small number of states of the channel is considered. Further detail is lost, but computational efficiency gained, when one takes the limit of a large number of ion channels, arriving at a deterministic kinetic model for the macroscopic ion channel current. Fluctuations of the current due to the stochastic opening and closing of individual ion channels can on the one hand be physiologically relevant, and on the other hand they can be exploited to infer single-channel properties, even when the single-channel current is too small to be measured directly (fluctuation analysis). In computational models of neurons, one can try to reproduce known cellular processes on the basis of measured properties of the microscopic constituents of the system.

Studying the Fisher zeros of the corresponding generalized “partition function,” the nonanalyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs) is probed. In contrast to the sudden quenching case, it is observed that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting “lobe” structure of Fisher zeros emerge. A connection to the topological phase transition is also made by studying the dynamical topological order parameter as a function of time ($t$) measured from the instant when the quenching is complete.

We present a method based on graph theory for evaluation of the inelastic propensity rules for molecules exhibiting complete destructive quantum interference in their elastic transmission. The method uses an extended adjacency matrix corresponding to the structural graph of the molecule for calculating the Green function between the sites with attached electrodes and consequently states the corresponding conditions the electron-vibration coupling matrix must meet for the observation of an inelastic signal between the terminals. The method can be fully automated and we provide a functional website running a code using Wolfram Mathematica, which returns a graphical depiction of destructive quantum interference configurations together with the associated inelastic propensity rules for a wide class of molecules.

We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice (Cayley tree) and Random Regular Graphs (RRG). In the latter case (RRG), the problem can be described as a tight-binding model on a lattice with N sites that is only locally a tree with constant connectivity. In certain sense, the RRG ensemble can be seen as infinite-dimensional cousin of Anderson model in d dimensions. For Bethe lattice [1], we show that the wave function amplitude at the root of a tree is distributed fractally in a large part of the delocalized phase. The fractal exponents are expressed in terms of the decay rate and the velocity in a problem of propagation of a front between unstable and stable phases. We demonstrate a crucial difference between a loopless Cayley tree and a locally tree-like structure without a boundary (random regular graph) where extended wavefunctions are ergodic. For RRG [2], we focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that presence of even very large loops (with size comparable with the diameter of the structure) modifies the properties of delocalized phase a lot. We show that the numerics can be interpreted in terms of the finite-size crossover from small to large system, separated by the correlation volume $N_c$, diverging exponentially at the localization transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wavefunction statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly non-trivial. Our results support an analytical prediction that states in the delocalized phase (and at $N \gg N_c$) are ergodic.

Driving a system into a steady state away from equilibrium, Bose condensation can occur also in an excited single-particle state or even several condensates can form (fragmented condensation). Examples for such behavior are found in exciton-polariton systems, which are driven-dissipative due to the interplay between particle influx from a pumped reservoir on the one hand and particle loss on the other. Here we present a theory for such non-equilibrium Bose condensation. Our analysis is based on a kinetic rate equation for a gas of non-interacting quasiparticles subjected to loss and in contact both with a heat bath as well as with a pumped particle reservoir. We show that with increasing pump power generically the system undergoes a sequence of transitions. Each transition involves a single-particle state that either starts or ceases to host a Bose condensate. While the first transition resembles lasing, in the limit of very strong pumping Bose condensation is determined by the coupling to the heat bath only, so that a single condensate in the single-particle ground state emerge like in equilibrium. For intermediate pumping strengths situations with several condensates have to occur.