Advances in experimental techniques have opened new routes for harnessing non-equilibrium dynamics in mesoscopic quantum systems. In this context, we study the impact of on-site interactions on the transport properties of a continuous quantum thermal machine composed of two coupled oscillators connected to two thermal reservoirs. In the weak system-reservoir coupling regime, where a long-standing debate concerns which reduced description should be preferred, we first show that the Redfield master equation (RME) provides an accurate and unifying framework that interpolates between two well-known limits: the \textit{local} and \textit{global} master equations. By relying on the Hierarchy of Pure States (HOPS), a numerically exact stochastic method, we then explore the full parameter space and show that interactions can be leveraged to tune the efficiency of the thermal machine at high temperatures (while leaving it essentially unchanged at low temperatures), induce non-reciprocal transport under asymmetric reservoir couplings, and generate steady-state entanglement within the junction. We derive expressions for system-bath correlators, such as heat and particle currents, consistently across different frameworks. Our work features on-site interactions to enhance the versatility of quantum thermodynamic junctions and clarifies the role of non-Markovianity and non-linearities in quantum transport.

Ultracold atoms in multimode optical cavities provide a rich testbed for many-body phenomena enabled by light-mediated interactions. Recent experiments include realizations of spin glasses and associative memories, as described by multimode Dicke models with disordered couplings. However, the properties of multimode Dicke models with ordered coupling geometries remain largely unexplored. In this work, we investigate the stable steady-states of the multimode Dicke model with an ordered nearest-neighbor coupling geometry, where $n_c$ atomic clusters are coupled via $n_c-1$ cavity modes. We show that the number of mean-field stable steady-states in the superradiant phase exhibits Fibonacci scaling with the number of atomic clusters, and that a subset of these steady-states exhibit persistent oscillations. Using both the truncated Wigner approximation and the numerically-exact hierarchy of pure states, we further demonstrate that these features of the stable steady-state solutions persist for finite cluster sizes. Ordered multimode Dicke models, such as the nearest-neighbor coupling geometry considered here, are accessible with current experimental technologies and point toward a broader class of strongly interacting dissipative systems with similarly rich behavior.

It is known that Green's functions can be expressed as continued fractions; the content at the $n$-th level of the fraction is encoded in a coefficient $b_n$, which can be recursively obtained using the Lanczos algorithm. We present a theory concerning errors in approximating Green's functions using continued fractions when only the first $N$ coefficients are known exactly. Our focus lies on the stitching approximation (also known as the recursion method), wherein truncated continued fractions are completed with a sequence of coefficients for which exact solutions are available. We assume a now standard conjecture about the growth of the Lanczos coefficients in chaotic many-body systems, and that the stitching approximation converges to the correct answer. Given these assumptions, we show that the rate of convergence of the stitching approximation to a Green's function depends strongly on the decay of staggered subleading terms in the Lanczos cofficients. Typically, the decay of the error term ranges from $1/\mathrm{poly}(N)$ in the best case to $1/\mathrm{poly}(\log N)$ in the worst case, depending on the differentiability of the spectral function at the origin. We present different variants of this error estimate for different asymptotic behaviours of the $b_n$, and we also conjecture a relationship between the asymptotic behavior of the $b_n$'s and the smoothness of the Green's function. Lastly, with the above assumptions, we prove a formula linking the spectral function's value at the origin to a product of continued fraction coefficients, which we then apply to estimate the diffusion constant in the mixed field Ising model.

The high complexity of many-body quantum dynamics means that essentially all approaches either exploit special structure or are approximate in nature. One such approach--the memory function formalism--involves a carefully chosen split into fast and slow modes. An approximate model for the fast modes can then be used to solve for Green's functions $G(z)$ of the slow modes. Using a formulation in operator Krylov space known as the recursion method, we prove the emergence of a universal random matrix description of the fast mode dynamics. This is captured by the level-$n$ Green's function $G_n(z)$, which we show approaches universal scaling forms in the fast limit $n\to \mathrm{\infty }$. Notably, this emergent universality can occur in both chaotic and non-chaotic systems, provided their spectral functions are smooth. This universality of $G_n(z)$ is precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is no explicit randomness present in the Hamiltonian. At finite $z$ we show that $G_n(z)$ approaches the Wigner semicircle law, while if $G(z)$ is the Green's function of certain hydrodynamical variables, we show that at low frequencies $G_n(z)$ is instead governed by the Bessel universality class from RMT. As an application of this universality, we give a numerical method--the spectral bootstrap--for approximating spectral functions from Lanczos coefficients. Our proof involves a map to a Riemann-Hilbert problem which we solve using a steepest-descent-type method, rigorously controlled in the $n\to \mathrm{\infty }$ limit. We are led via steepest-descent to a Coulomb gas optimization problem, and we discuss how a recent conjecture--the `Operator Growth Hypothesis--is linked to a confinement transition in this Coulomb gas. These results elevate the recursion method to a theoretically principled framework with universal content.

Using artificial dissipation to tame entanglement growth, we chart the emergence of diffusion in a generic interacting lattice model for varying U(1) charge densities. We follow the crossover from ballistic to diffusive transport above a scale set by the scattering length, finding the intuitive result that the diffusion constant scales as $D\propto 1/\rho$ at low densities $\rho$. Our numerical approach generalizes the Dissipation-Assisted Operator Evolution (DAOE) algorithm: in the spirit of the BBGKY hierarchy, we effectively approximate non-local operators by their ensemble averages, rather than discarding them entirely. This greatly reduces the operator entanglement entropy, while still giving accurate predictions for diffusion constants across all density scales. We further construct a minimal model for the transport crossover, yielding charge correlation functions which agree well with our numerical data. Our results clarify the dominant contributions to hydrodynamic correlation functions of conserved densities, and serve as a guide for generalizations to low temperature transport.

Scrambling of quantum information in unitary evolution can be hindered due to measurements and localization, which pin quantum mechanical wavefunctions in real space suppressing entanglement in the steady state. In monitored free-fermionic models, the steady state undergoes an entanglement transition from a logarithmically entangled critical state to area-law. However, disorder can lead to Anderson localization. We investigate free fermions in a random potential with continuous monitoring, which enables us to probe the interplay between measurement-induced and localized phases. We show that the critical phase is stable up to a finite disorder and the criticality is consistent with the Berezinskii-Kosterlitz-Thouless universality. Furthermore, monitoring destroys localization, and the area-law phase at weak dissipation exhibits power-law decay of single-particle wave functions. Our work opens the avenue to probe this novel phase transition in electronic systems of quantum dot arrays and nanowires, and allow quantum control of entangled states.

Conservation laws can constrain entanglement dynamics in isolated quantum systems, manifest in a slowdown of higher R\'enyi entropies. Here, we explore this phenomenon in a class of long-range random Clifford circuits with U$(1)$ symmetry where transport can be tuned from diffusive to superdiffusive. We unveil that the different hydrodynamic regimes reflect themselves in the asymptotic entanglement growth according to $S(t)\propto t^{1/z}$, where the dynamical transport exponent $z$ depends on the probability $\propto r^{-\alpha }$ of gates spanning a distance $r$. For sufficiently small $\alpha$, we show that the presence of hydrodynamic modes becomes irrelevant such that $S(t)$ behaves similarly in circuits with and without conservation law. We explain our findings in terms of the inhibited operator spreading in U$(1)$-symmetric Clifford circuits, where the emerging light cones can be understood in the context of classical L\'evy flights. Our work sheds light on the connections between Clifford circuits and more generic many-body quantum dynamics.

Many-body quantum systems are notoriously hard to study theoretically due to the exponential growth of their Hilbert space. It is also challenging to probe the quantum correlations in many-body states in experiments due to their sensitivity to external noise. Using synthetic quantum matter to simulate quantum systems has opened new ways of probing quantum many-body systems with unprecedented control, and of engineering phases of matter which are otherwise hard to find in nature. Noisy quantum circuits have become an important cornerstone of our understanding of quantum many-body dynamics. In particular, random circuits act as minimally structured toy models for chaotic nonintegrable quantum systems, faithfully reproducing some of their universal properties. Crucially, in contrast to the full microscopic model, random circuits can be analytically tractable under a reasonable set of assumptions, thereby providing invaluable insights into questions which might be out of reach even for state-of-the-art numerical techniques. Here, we give an overview of two classes of dynamics studied using random-circuit models, with a particular focus on the dynamics of quantum entanglement. We will especially pay attention to potential near-term applications of random-circuit models on noisy-intermediate scale quantum (NISQ) devices. In this context, we cover hybrid circuits consisting of unitary gates interspersed with nonunitary projective measurements, hosting an entanglement phase transition from a volume-law to an area-law phase of the steady-state entanglement. Moreover, we consider random-circuit sampling experiments and discuss the usefulness of random quantum states for simulating quantum many-body dynamics on NISQ devices by leveraging the concept of quantum typicality. We highlight how emergent hydrodynamics can be studied by utilizing random quantum states generated by chaotic circuits.

Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume and area-law entanglement. There is a correspondence between measurement-induced transitions in non-unitary quantum circuits in $d$ spatial dimensions and classical statistical mechanical models in $d+1$ dimensions. In certain limits these models map to percolation, but there is analytical and numerical evidence to suggest that away from these limits the universality class should generically be distinct from percolation. Intriguingly, despite these arguments, numerics on 1D qubit circuits give bulk exponents which are nonetheless close to those of 2D percolation, with possible differences in surface behavior. In the first part of this work we study the critical properties of 2D Clifford circuits. In the bulk, we find many properties suggested by the percolation picture, including matching bulk exponents, and an inverse power-law for the critical entanglement growth, $S(t,L)\sim L(1-a/t)$, which saturates to an area-law. We then utilize a graph-state based algorithm to analyze in 1D and 2D the critical properties of entanglement clusters in the steady state. We show that in a model with a simple geometric map to percolation, the projective transverse field Ising model, the entanglement clusters are governed by percolation surface exponents. However, in the Clifford models we find large deviations in the cluster exponents from those of surface percolation, highlighting the breakdown of any possible geometric map to percolation. Given the evidence for deviations from the percolation universality class, our results raise the question of why nonetheless many bulk properties behave similarly to percolation.

The resilience of quantum entanglement to a classicality-inducing environment is tied to fundamental aspects of quantum many-body systems. The dynamics of entanglement has recently been studied in the context of measurement-induced entanglement transitions, where the steady-state entanglement collapses from a volume-law to an area-law at a critical measurement probability $p_c$. Interestingly, there is a distinction in the value of $p_c$ depending on how well the underlying unitary dynamics scramble quantum information. For strongly chaotic systems, $p_c>0$, whereas for weakly chaotic systems, such as integrable models, $p_c=0$. In this work, we investigate these measurement-induced entanglement transitions in a system where the underlying unitary dynamics are many-body localized (MBL). We demonstrate that the emergent integrability in an MBL system implies a qualitative difference in the nature of the measurement-induced transition depending on the measurement basis, with $p_c>0$ when the measurement basis is scrambled and $p_c=0$ when it is not. This feature is not found in Haar-random circuit models, where all local operators are scrambled in time. When the transition occurs at $p_c>0$, we use finite-size scaling to obtain the critical exponent $\nu =1.3(2)$, close to the value for 2+0D percolation. We also find a dynamical critical exponent of $z=0.98(4)$ and logarithmic scaling of the R\'{e}nyi entropies at criticality, suggesting an underlying conformal symmetry at the critical point. This work further demonstrates how the nature of the measurement-induced entanglement transition depends on the scrambling nature of the underlying unitary dynamics. This leads to further questions on the control and simulation of entangled quantum states by measurements in open quantum systems.

Periodically driven quantum systems host a range of non-equilibrium phenomena which are unrealizable at equilibrium. Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, a putative quantum phase of matter. We present the observation of discrete time crystalline order in a driven system of paramagnetic $P$ -donor impurities in isotopically enriched ${}^{28}Si$ cooled below $10$ K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of DTC order. We propose a theoretical model based on the paradigmatic central spin model which is in good agreement with experimental observations, and investigate the role of dissipation in the stabilisation of the DTC. Both experiment and theory indicate that the order in this system is primarily a dissipative effect, and which persists in the presence of spin-spin interactions. We present a theoretical phase diagram as a function of interactions and dissipation for the central spin model which is consistent with the experiments. This opens up questions about the interplay of coherent interaction and dissipation for time-translation symmetry breaking in many-body Floquet systems.