Understanding and probing phase transitions in non-equilibrium systems is an ongoing challenge in physics. A particular instance are phase transitions that occur between a non-fluctuating absorbing phase, e.g., an extinct population, and one in which the relevant order parameter, such as the population density, assumes a finite value. Here we report the observation of signatures of such a non-equilibrium phase transition in an open driven **quantum** system. In our experiment rubidium atoms in a quasi one-dimensional cold disordered gas are laser-excited to Rydberg states under so- called facilitation conditions. This conditional excitation process competes with spontaneous decay and leads to a crossover between a stationary state with no excitations and one with a finite number of excitations. We relate the underlying physics to that of an absorbing state phase transition in the presence of a field (i.e. off-resonant excitation processes) which slightly offsets the system from criticality. We observe a characteristic power-law scaling of the Rydberg excitation density as well as increased fluctuations close to the transition point. Furthermore, we argue that the observed transition relies on the presence of atomic motion which introduces annealed disorder into the system and enables the formation of long-ranged correlations. Our study paves the road for future investigations into the largely unexplored physics of non-equilibrium phase transitions in open **many**-**body** **quantum** systems.

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Modeling **many**-**body** **quantum** systems with strong interactions is one of the core challenges of modern physics. A range of methods has been developed to approach this task, each with its own idiosyncrasies, approximations, and realm of applicability. However, there remain **many** problems that are intractable for existing methods. In particular, **many** approaches face a huge computational barrier when modeling large numbers of coupled electrons and ions at finite temperature. Here, we address this shortfall with a new approach to modeling **many**-**body** **quantum** systems. On the basis of the Bohmian trajectory formalism, our new method treats the full particle dynamics with a considerable increase in computational speed. As a result, we are able to perform large-scale simulations of coupled electron-ion systems without using the adiabatic Born-Oppenheimer approximation.

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At the same time, this is also a relatively new subfield with **many** open questions, ranging from specific questions such as the effect of spontaneous emissions on dynamical processes used to prepare states with cold atoms in optical lattices, to broader questions such as to what extent and under which conditions it is possible to identify universality classes for the steady-state of driven, dissipative **many**-**body** systems. While this review has focussed on markovian dissipative systems, which are prevalent in **quantum** optics and AMO implementations, one important area for future development will be the further development of these ideas in non-markovian regimes. Already there has been a great deal of progress on treating non- markovian effects in open systems [1, 355], including through the extension of master equation techniques [356, 357] and stochastic wavefunction methods [358–360] on expanded Hilbert spaces. In terms of **quantum** trajectories methods, there have been various extensions to the method to include memory effects of the reservoir for single-particle systems, beginning with work by Imamoglu in Ref. [361], which involved the coupling of a small **quantum** system to a damped cavity with a finite lifetime, and leading up to recent direct extensions to **quantum** trajectories methods on the Hilbert space of the reduced system [362, 363]. Recent studies have started combining this with Bose-Einstein condensates, including work where atoms are used as a probe immersed in the condensate [364, 365]. Combined with strongly interacting systems, such ideas may open new pathways for investigations of fundamental dynamics, or provide practical tools for out- of-equilibrium preparation of **many**-**body** states as discussed in section V. Extensions to non-markovian systems may also bring new opportunities for the study of solid-state systems via these techniques [1].

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This can easily be seen to be a linear operator from Hilbert space H d ⊗ N to H d ⊗N . Such operators are called Matrix Product Operators (MPO), because they are described as a product of matrices M i, j . We will come across various applications of MPOs in the course of this thesis. As one can see, MPO representation can produce highly entangled, non-local operator very easily. As we will see, MPOs provide a very important formalism for the study of TN representation of topological states in 2D. The matrix product formalism [26, 27] has played a significant role in the study of one dimensional systems. In particular, the matrix product representation of 1D **quantum** states underlies successful numerical algorithms like the Density Matrix Renormalization Group algorithm [28] and the Time-Evolving Block Decimation algorithm [29]. Moreover, the matrix product representation provides a deep insight into the structure of the ground states in 1D [27], which enables rigorous proofs of the efficiency of 1D variational algorithms in search for the ground states [30, 31] and also a complete classification of 1D gapped phases [25, 32–34].

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are suppressed compared with the expectations for well-bound nuclei. This general observation was later supported by theoretical calculations [24], and later this e ff ect was seen in **many** other reactions using weakly bound stable [25–36] as well as unstable beams [22]. Answering whether fusion cross- sections below the barrier are suppressed or enhanced compared with well-bound nuclei is however more fraught, since enhancement due to channel couplings and e ff ects of breakup are both important. A clear answer requires model calculations and current models are not quantitative enough to obtain dependable predictions. Quantitative calculations firstly require a complete understanding and then modelling of the mechanisms leading to breakup. Understanding breakup mechanisms, particularly of α-cluster nuclei, has had a long history. The most obvious breakup mechanism, i.e. resulting from excitation of states (or resonances) above the breakup threshold, is well documented [37, 38]. In addi- tion to this mechanism, breakup following transfer of nucleons had also been observed [38, 39]. Such a mechanism is likely to be significant in low energy nuclear reactions involving halo nuclei in partic- ular [40–42]. Breakup triggered by transfer has been shown to be important for weakly-bound stable beams. Recent experiments have provided a complete picture of all the physical mechanisms that trig- ger breakup [43–45], as well as important information on breakup time-scales. These experiments, described next, provide clear insights for incorporation into realistic models.

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Research into mixed-state time evolution and Lindblad dynamics has also pro- gressed. It has been proven that density matrices (and purifications) of Gibbs states of local Hamiltonians have efficient matrix product representations [85, 154, 163, 247]. Two schools of thought have used this insight to develop a series of methods for simulating time evolution. One school employs density matrices [21, 41, 95, 95, 101, 151, 152, 167, 168, 175–178, 212, 221, 222, 247]. They note that the space of operators on a spin chain is the tensor product of onsite operator spaces, just as the space of **many**-**body** pure states being a tensor product on onsite Hilbert spaces; the chief difference (in this view) is merely the dimensionality of the onsite space. For example, on a spin-half chain, the space of onsite operators is four-dimensional, while the space of pure states is two dimensional. This school then applies familiar pure state methods, including the creation and truncation of matrix product states and time evolution by TEBD, to density matrices—which are, after all, vectors in a larger space. The resulting truncation algorithms minimize the error ac- cording to the Hilbert-Schmidt (Frobenius) norm. In certain situations—in particular, dynamics near thermal equilibrium or a non-equilibrium steady state—this approach works well. In other situations, however—in particular, time evolution starting from a pure state—the density matrices suffer from a catastrophic loss of positivity. (Even checking positivity is NP-hard in the system size [117].)

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We prove that Gibbs measures of nonlinear Schr¨ odinger equations arise as high-temperature limits of thermal states in **many**-**body** **quantum** mechanics. Our results hold for defocusing inter- actions in dimensions d = 1, 2, 3. The **many**-**body** **quantum** thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for d = 2, 3. In dimensions d = 2, 3, the Gibbs measures are supported on singular distri- butions, and a renormalization of the chemical potential is necessary. On the **many**-**body** **quantum** side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original **many**-**body** **quantum** problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

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We theoretically investigate the **many**-**body** states of exciton-polaritons that can be observed by pump-probe spectroscopy in high-Q inorganic microcavities. Here, a weak-probe “spin-down” polariton is introduced into a coherent state of “spin-up” polaritons created by a strong pump. We show that the ↓ impurities become dressed by excitations of the ↑ medium, and form new polaronic quasiparticles that feature two-point and three-point **many**-**body** **quantum** correlations, which, in the low density regime, arise from coupling to the vacuum biexciton and triexciton states respectively. In particular, we find that these correlations generate additional branches and avoided crossings in the ↓ optical transmission spectrum that have a characteristic dependence on the ↑-polariton density. Our results thus demonstrate a way to directly observe correlated **many**-**body** states in an exciton-polariton system that go beyond classical mean-field theories.

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I would like to thank also all the nice people I have met, collaborate and spent time together in the **many** visits to the US in this three years. Start- ing from south–west and making a clock–wise round I start from the people in Los Alamos, Joe Carlson and Stefano Gandolfi for the great hospitality and the **many** interesting discussions. In the Bay Area my biggest thank is for Mal Kalos, not only for having patiently guided me through the mysteries of Fermionic Monte Carlo, but also for the hospitality and for his knowledgeable and passionate introduction to Asian cuisine. My grateful thanks are specially for Eric Schwegler and the guys at LLC for allowing the use of their computa- tional facilities on which most of the research presented in this work have been carried out. Thank you also to Jonathan and Ethan for the nice time and the **many** interesting discussions. Going further north in the sunny Seattle I wish to thank the people in UW/INT, especially Sanjay Reddy, Jeremy and Ermal for the hospitality and for the **many** stimulating interactions during my visits. A special thank is due to Ermal for the good time we had together and for his help. Changing coast and going to Ithaca I would like to thank Cyrus Umrigar, for the exquisite hospitality in my visits to Cornell and for having taught me so **many** things about Monte Carlo. I have enjoyed the time spent there, and this also thanks to Adam, Matt, John, Andreas and the beers/barbecue/whiskeys we had together.

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From the point of view of a committed **quantum** theorist, who knows neither of any experimental evidence for any breakdown in **quantum** theory, nor of any alternative theory which is not both ad hoc and incompatible with special relativity, these asser- tions may seem plausible (cf. Deutsch 1996). It is certainly the case that, at least over short time intervals, **quantum** states can be found which will give apparently accu- rate representations of the physical states of essentially all non-gravitational physical systems, however large or complex they may be. Indeed, states can be found which are not only “apparently accurate” in the sense that they are compatible with all our actual short-term observations, but also in the sense that they represent any parti- cle as being, during the interval considered, sufficiently well-localized to accord with conventional pictures of that type of **quantum** particle. Thus, in such states, the atoms in our environment are represented as slightly fuzzy balls, while free electrons are represented as moving along slightly fuzzy straight lines. I shall call these states “pragmatic”. For example, a pragmatic state for a gas might use minimum uncer- tainty wave-packets centred on definite choices of positions and momenta to describe the centre of mass variables of the molecules, together with appropriate choices of molecular wavefunctions for the electronic variables. At a more sophisticated level, using our excellent understanding of **quantum** states for single molecules, and the possibility of building such **quantum** states up to describe **many** molecules, pragmatic **quantum** states can be ascribed at any time to any chemical system – including the human brain. The fundamental problem – the problem of “Schr¨ odinger’s cat” – arises because the time continuation given by the **quantum** mechanical dynamics does not always lead from a **quantum** state which provides an apparently accurate description at one time to a **quantum** state which provides a similarly apparently accurate de- scription at later times. The pragmatism of a state, in other words, is time dependent. Consider, for example, electrons contributing to the production of an interference pattern by passing, one at a time, through some kind of two-slit device and hitting a position detector. (Pictures from such an experiment are presented in Tonomura et al. 1989.) I have chosen this example, partly because the pictures are such a direct demonstration of the difference between **quantum** state and observation, partly because the electrons are more likely to be seen to hit some parts of the detector than others, and partly because I think that the two-alternative experiments by which this subject is usually introduced foster a na¨ıve view of the complexity of **quantum** states (cf. Weinstein 1996).

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However, recent advances in Cloud Service Provider (CSP) technologies have narrowed the per- formance gap vis-a-vis supercomputers. Clusters of compute nodes with faster interconnects are now offered. Instances with physical memory on the order of hundreds of GiB are now provided. Faster processors and coprocessors are even in development for new instance types. For exam- ple, the Amazon Elastic Compute Cluster (EC2) has provided access to Graphics Processing Unit (GPU) instance types, which are optimal for linear algebra operations, and Google has pioneered Domain Specific Architectures (DSAs) with the creation and provision of Tensor Processing Units (TPUs), which are optimal for the **many** small matrix-matrix operations needed in deep learning applications. While these advances are significant, there is still a challenge in managing high-order tensor contractions in QMB applications. This is because both the number of operations and the memory footprint involved in the computation grow exponentially with the system size.

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We consider spinless fermions on a finite one-dimensional lattice, interacting via nearest-neighbor repulsion and subject to a strong electric field. In the noninteracting case, due to Wannier-Stark localization, the single-particle wave functions are exponentially localized even though the model has no quenched disorder. We show that this system remains localized in the presence of interactions and exhibits physics analogous to models of conventional **many**-**body** localization (MBL). In particular, the entanglement entropy grows logarithmically with time after a quench, albeit with a slightly different functional form from the MBL case, and the level statistics of the **many**-**body** energy spectrum are Poissonian. We moreover predict that a quench experiment starting from a charge-density wave state would show results similar to those of Schreiber et al. [Science 349 , 842 (2015)].

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Consider an electron travelling a QPC via the lowest 1D energy subband. The electron can exist there either in the state of superposition or—if it is being observed—in one of the possible spin states; the latter excludes the interference. Most importantly, it is not necessary to perform any real observation of the electron to suppress the interference. As demonstrated in the double-slit experiments, it is enough to create experimental conditions al- lowing such observation (see e.g. [18]), which is one of mysteries of **quantum** mechanics.

Our results underline the utility of cold atoms in optical lattices for the investigation of fundamental physical processes driven by small-amplitude terms and specifically higher-order tunneling. By partly freezing the motion in the deep lattice, these sensitive processes can be observed here despite finite initial temperatures (which lead to defects and missing atoms). This will motivate further investigation of **quantum** phases and critical properties near these higher- order resonances, which are presently unknown, including systems with tilts along multiple axes (19, 31). Our initial studies of parameter reversals also open the door to the study of **many**-**body** dephasing and echo-type experiments on a **quantum** **many**-**body** system, as well as investigations into the nature of the **many**-**body** dephasing and (apparent) thermalization (32). Parallels can be drawn with arrays of **quantum** dots, opening further possibilities to model electron tunneling over multiple sites (11) by using fermionic atoms.

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DOI: 10.4236/ampc.2018.81003 33 Advances in Materials Physics and Chemistry logical properties that are currently under intense investigation. When certain types of such materials ( i.e. 2D Topological Insulators) are subject to a perpen- dicular magnetic field, they may as well undergo a phase transition to **Quantum** Hall Insulators [4], violating the time reversal symmetry that controls the topol- ogy of the surface states. Normally, there is a transverse (to B) small electric field E, which—to first order in E—is responsible for the macroscopic quantization of the Hall conductivity [5], and which is the central quantity in the present paper. Interestingly enough, the strong E-field regime has not been investigated in suf- ficient detail so far, in particular with respect to the role of the E-field on ther- modynamic **many**-**body** properties (see however [6], and for some earlier attempts see [7]-[14]), as these properties are determined in the noninteracting electrons framework (the one that, in any case, pertains to the Integer Hall Effect regime). In this work, we present potential consequences (on thermodynamic and trans- port properties) of a strong electric field applied tangentially to a macroscopic 2D graphene sheet, when also subjected to a perpendicular magnetic field of ar- bitrary strength.

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In section IV, we saw that the minimum-error and the minimum Bayes cost comparison strategies always are projective measurements in some basis. Thus these comparison strategies can always be realised in principle, but as for the universal error-free comparison, it may require a measurement in a highly entangled basis, making the realisation difficult in practice. We gave an example of a comparison strategy for determining, with minimum error, whether the states of two **quantum** systems were identical or not. The states of the each of the systems were known to be in one of three given **quantum** states. In the particular case we considered, the **quantum** comparison can be realised with a beam splitter in a way similar to the universal error-free comparison. We need to distinguish the state 1/ √ 2( | + + i + | − −i ) from the other Bell states, as this state corresponds to the outcome ‘same’ and the other Bell states (or linear combinations of these) correspond to the outcome ‘different’. One way to do this is to apply a rotation of π/2 to one of the **quantum** systems, so that | + i ↔ |−i , and 1/ √ 2( | + + i + | − −i ) transforms into 1/ √ 2( | + −i + | − + i ). If two **quantum** systems in this state are incident on a beam splitter, both systems will exit together since the state is symmetric. But this case can readily be distinguished from the other symmetric states, 1/ √ 2( | + + i ± | − −i ), since it is the only one where the **quantum** systems have different states when each of them is measured in the basis {| + i , |−i} . To do this, direct each of the beam splitter outputs onto a polarising beam splitter, separating | + i and |−i , and look for clicks at both outputs of one of the polarising beam splitters. Note that since the comparison situation is not invariant under unitary transforms, we need to know in which basis to perform the measurement. Essentially, this method has already been used to individually distinguish two Bell states from each other and from the other two Bell states [11].

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So Everett’s view is that the deterministic Schroedinger equation is always right, in the sense that the **quantum** state of an isolated system always evolves in accordance with it. And the **quantum** state ‘is everything’ in the sense that values are assigned to physical quantities only by the orthodox rules. In particular, no quantity is preferred by being assigned, in every state, a value; as is proposed, for the quantity position, by the pilot-wave approach. But to reconcile this uncollapsed and un-supplemented **quantum** state with the apparent fact that any **quantum** experiment has a single outcome, Everett then identifies the Appearances—our apparent macroscopic realm, with its various experiments’ outcomes---with one of a vast multiplicity of realms. These are often called ‘branches’ rather than ‘worlds’.

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This use of the reciprocal theorem allows any swimmer problem to be solved, provided the appropriate conjugate stress tensor is known. Fortunately **many** exact and ap- proximate solutions of Stokes ﬂow problems for a variety of geometries exist in the literature, so that the relevant stress tensor is available in **many** cases. As we have seen, the general three-dimensional case is easily solved using an approximate integration kernel, giving good asymptotic results. However, this approximation is not appropriate in the near ﬁeld, where the separation becomes compara- ble to the swimmer size. We will now use a classic two- dimensional exact result for the stress tensor to shed light on these cases.

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Literature The central concept of this work is **Many**-**Body** Localization (MBL) - a property of **quantum**-mechanical **many**-**body** systems. The original work (An- derson, 1958) posed a question whether the MBL can exist and as a way to solve it studied the localization of a single particle. The distinction between the MBL and the Anderson localization is not strict: a state of an L-spin system can be represented by a position of a single particle in an L-dimensional hypercube (Thouless, 1977). The first MBL system with a mathematical solution converging in a realistic range of parameters (such as the interaction strength, lattice connectivity and temperature) was presented by (Basko, Aleiner, and B.L. Altshuler, 2006). The system they con- sidered contained interacting 2d electrons in a strongly disordered material (grains of metal with a weak tunneling between them). They have proved zero conductance at finite temperature as an experimental manifestation of MBL. The idea was univer- sal enough to be applicable to other **many**-**body** systems. The simplification to spin chains soon followed: strongly disordered spin chains have been claimed to possess MBL, which manifested in rich mathematical structure including zero conductance. The intuition for said mathematical structure was pioneered by conjectures of local conserved quantities (Oganesyan and Huse, 2007; Serbyn, Papić, and Abanin, 2013) and finite depth local circuits (Bauer and Nayak, 2013). The latter conjectured that an MBL spin chain Hamiltonian could be diagonalized by such circuit, at least approximately. It culminated in an exact construction by (Imbrie, 2016): a proof that a certain disordered spin chain can be solved exactly if we extend our notion of finite depth local circuits slightly. We will use that construction to obtain a better understanding of MBL phase.

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