The global coupling of few-level quantum systems (``spins'') to a discrete set of bosonic modes is a key ingredient for many applications in quantum science, including large-scale entanglement generation, quantum simulation of the dynamics of long-range interacting spin models, and hybrid platforms for force and spin sensing. We present a general numerical framework for treating the out-of-equilibrium dynamics of such models based on matrix product states. Our approach applies for generic spin-boson systems: it treats any spatial and operator dependence of the two-body spin-boson coupling and places no restrictions on relative energy scales. We show that the full counting statistics of collective spin measurements and infidelity of quantum simulation due to spin-boson entanglement, both of which are difficult to obtain by other techniques, are readily calculable in our approach. We benchmark our method using a recently developed exact solution for a particular spin-boson coupling relevant to trapped ion quantum simulators. Finally, we show how decoherence can be incorporated within our framework using the method of quantum trajectories, and study the dynamics of an open-system spin-boson model with spatially nonuniform spin-boson coupling relevant for trapped atomic ion crystals in the presence of molecular ion impurities.
The global coupling of few-level quantum systems (``spins'') to a discrete set of bosonic modes is a key ingredient for many applications in quantum science, including large-scale entanglement generation, quantum simulation of the dynamics of long-range interacting spin models, and hybrid platforms for force and spin sensing. We present a general numerical framework for treating the out-of-equilibrium dynamics of such models based on matrix product states. Our approach applies for generic spin-boson systems: it treats any spatial and operator dependence of the two-body spin-boson coupling and places no restrictions on relative energy scales. We show that the full counting statistics of collective spin measurements and infidelity of quantum simulation due to spin-boson entanglement, both of which are difficult to obtain by other techniques, are readily calculable in our approach. We benchmark our method using a recently developed exact solution for a particular spin-boson coupling relevant to trapped ion quantum simulators. Finally, we show how decoherence can be incorporated within our framework using the method of quantum trajectories, and study the dynamics of an open-system spin-boson model with spatially nonuniform spin-boson coupling relevant for trapped atomic ion crystals in the presence of molecular ion impurities.
Spin-boson models involving many interacting spins and bosons are ubiquitous in quantum simulation platforms. At the same time, characterizing the dynamics of these quantum systems represents a significant challenge. Here, we consider general spin-boson models where bosons are subject to Markovian dissipation (e.g., due to cavity loss). We present an exact hybrid quantum-classical stochastic approach where the solution of a classical stochastic equation—mimicking the bosonic modes—is input into a quantum stochastic equation for the spins. Furthermore, the spins are effectively decoupled for each stochastic realization, which nevertheless comes at the expense of sampling over unphysical states. In contrast with existing stochastic approaches based on the influence functional formalism, we place no restriction (factorizability or Gaussianity) on the initial state, or the spin-boson coupling (except that it be linear in the bosonic operator). Markovian dissipation, being at the heart of our approach, renders the stochastic equations Markovian even in the strong coupling regime. Furthermore, it ensures hermiticity (though not positivity) of the density matrix for each realization, thus improving the convergence of stochastic sampling. Interestingly, we find a condition on the classical simulability of the system based solely on the single atom cooperativity even in a many-body setting. We benchmark and showcase the utility of our approach in several examples, specifically in cases where a direct numerical computation is unfeasible.
Experimental advances have made it possible to realize and control quantum many-body systems, allowing the experimental study of non-equilibrium phenomena. Due to the Hilbert space growing exponentially with the number of particles, advanced numerical techniques are usually required to understand many-body effects at the theoretical level. In this thesis, we address the question of how to efficiently simulate the non-equilibrium dynamics of many-body systems using matrix product states (MPS). In particular, we consider quantum systems in interaction with macroscopic environments, as they appear in Anderson impurity problems, quantum thermodynamics, or open quantum systems. We develop a low-entanglement representation of the environment with only short-range interactions, perfectly suitable for simulations with MPS. We further show that an interleaved ordering of tight-binding chains can significantly reduce the creation of entanglement, as opposed to an intuitive implementation of the Hamiltonian's geometry. Our approach allows long-time simulations and an analysis of the environmental dynamics, in which, after a sudden quench, we find clear signatures of many-body effects. We employ our techniques to compute spectral functions of impurity models, with possible applications to dynamical mean-field theory impurity solvers, and to calculate dissipation in the non-equilibrium Anderson model with explicit time-dependence, as relevant for ongoing experiments with oscillating tip atomic force microscopes.
Theoretical models of spins coupled to bosons provide a simple setting for studying a broad range of important phenomena in many-body physics, from virtually mediated interactions to decoherence and thermalization. In many atomic, molecular, and optical systems, such models also underlie the most successful attempts to engineer strong, long-ranged interactions for the purpose of entanglement generation. Especially when the coupling between the spins and bosons is strong, such that it cannot be treated perturbatively, the properties of such models are extremely challenging to calculate theoretically. Here, exact analytical expressions for nonequilibrium spin-spin correlation functions are derived for a specific model of spins coupled to bosons. The spatial structure of the coupling between spins and bosons is completely arbitrary, and thus the solution can be applied to systems in any number of dimensions. The explicit and nonperturbative inclusion of the bosons enables the study of entanglement generation (in the form of spin squeezing) even when the bosons are driven strongly and near resonantly, and thus provides a quantitative view of the breakdown of adiabatic elimination that inevitably occurs as one pushes towards the fastest entanglement generation possible. The solution also helps elucidate the effect of finite temperature on spin squeezing. The model considered is relevant to a variety of atomic, molecular, and optical systems, such as atoms in cavities or trapped ions. As an explicit example, the results are used to quantify phonon effects in trapped ion quantum simulators, which are expected to become increasingly important as these experiments push towards larger numbers of ions.
Simulating the non-perturbative and non-Markovian dynamics of open quantum systems is a very challenging many body problem, due to the need to evolve both the system and its environments on an equal footing. Tensor network and matrix product states (MPS) have emerged as powerful tools for open system models, but the numerical resources required to treat finite-temperature environments grow extremely rapidly and limit their applications. In this study we use time-dependent variational evolution of MPS to explore the striking theory of Tamascelli et al. (Phys. Rev. Lett. 2019, 123, 090402.) that shows how finite-temperature open dynamics can be obtained from zero temperature, i.e., pure wave function, simulations. Using this approach, we produce a benchmark dataset for the dynamics of the Ohmic spin-boson model across a wide range of coupling strengths and temperatures, and also present a detailed analysis of the numerical costs of simulating non-equilibrium steady states, such as those emerging from the non-perturbative coupling of a qubit to baths at different temperatures. Despite ever-growing resource requirements, we find that converged non-perturbative results can be obtained, and we discuss a number of recent ideas and numerical techniques that should allow wide application of MPS to complex open quantum systems.
Measurement-induced phases exhibit unconventional dynamics as emergent collective phenomena, yet their behavior in tailored interacting systems - crucial for quantum technologies - remains less understood. We develop a systematic toolbox to analyze monitored dynamics in long-range interacting systems, relevant to platforms like trapped ions and Rydberg atoms. Our method extends spin-wave theory to general dynamical generators at the quantum trajectory level, enabling access to a broader class of states than approaches based on density matrices. This allows efficient simulation of large-scale interacting spins and captures nonlinear dynamical features such as entanglement and trajectory correlations. We showcase the versatility of our framework by exploring entanglement phase transitions in a monitored spin system with power-law interactions in one and two dimensions, where the entanglement scaling changes from logarithm to volume law as the interaction range shortens, and by dwelling on how our method mitigates experimental post-selection challenges in detecting monitored quantum phases.
Trapped atomic ion crystals are a leading platform for quantum simulations of spin systems, with programmable and long-range spin-spin interactions mediated by excitations of phonons in the crystal. We describe a complementary approach for quantum simulations of bosonic systems using phonons in trapped-ion crystals, here mediated by excitations of the trapped-ion spins. The scheme enables a high degree of programability across a dense graph of bosonic couplings, utilizing long-lived collective phonon modes in a trapped-ion chain. As such, it is well suited for tackling hard problems such as boson sampling and simulations of long-range bosonic and spin-boson Hamiltonians.