We study the dynamical response of a system to a sudden change of the tuning parameter starting ͑or ending͒ at the quantum critical point. In particular, we analyze the scaling of the excitation probability, number of excited quasiparticles, heat and entropy with the quench amplitude, and the system size. We extend the analysis to quenches with arbitrary power law dependence on time of the tuning parameter, showing a close connection between the scaling behavior of these quantities with the singularities of the adiabatic susceptibilities of order m at the quantum critical point, where m is related to the power of the quench. Precisely for sudden quenches, the relevant susceptibility of the second order coincides with the fidelity susceptibility. We discuss the generalization of the scaling laws to the finite-temperature quenches and show that the statistics of the low-energy excitations becomes important. We illustrate the relevance of those results for cold-atom experiments.PACS number͑s͒: 64.70. Tg, 64.70.qj, 67.85.Ϫd Understanding the dynamics of quantum interacting systems is one of the key challenges of the modern physics. The theoretical research in this area has been stimulated by the fast developments in the field of cold-atom experiments 1 and the emerging possibility of manipulating quantum systems. 2 In particular, the idea of suddenly changing a parameter of the Hamiltonian, i.e., performing a quench, has been widely addressed within different approaches.3-6 Furthermore, recently the interest in studying quenches has been motivated by the questions on the thermalization of quantum systems. 7 Quenches near quantum critical points are especially interesting because of the expected universality of the response of the system and thus the possibility of using the quench dynamics as a nonequilibrium probe of phase transitions. This universality is well established in equilibrium systems. 8 It has been shown that a universal scaling arises as well in the case of slow adiabatic perturbations that drive the system through a quantum critical point. 9,10 The predicted scaling for the number of created quasiparticles with the quench rate was verified for various specific models. [11][12][13][14][15][16] This analysis was also generalized to nonlinear quenches. 17,18 In this work we consider a d-dimensional system described by a Hamiltonian H͑͒ = H 0 + V, where H 0 is the Hamiltonian corresponding to a quantum critical point ͑QCP͒ and V is a relevant ͑or marginal͒ perturbation, which drives the system to a particular phase. The quench process is implemented through the parameter that changes in time, according to some protocol, between the initial value =0 at time t = 0, corresponding to the critical point, and the final value f at final time t f . 29 We expect two qualitatively different scenarios: ͑i͒ for fast quenches, the response of the system only depends on the quench amplitude f , but not on the details of the protocol used to change -we refer to this regime as sudden quench; ͑ii͒ for slow quenches, i...

We propose a method to study dynamical response of a quantum system by evolving it with an imaginary-time dependent Hamiltonian. The leading non-adiabatic response of the system driven to a quantum-critical point is universal and characterized by the same exponents in real and imaginary time. For a linear quench protocol, the fidelity susceptibility and the geometric tensor naturally emerge in the response functions. Beyond linear response, we extend the finite-size scaling theory of quantum phase transitions to non-equilibrium setups. This allows, e.g., for studies of quantum phase transitions in systems of fixed finite size by monitoring expectation values as a function of the quench velocity. Non-equilibrium imaginary-time dynamics is also amenable to quantum Monte Carlo (QMC) simulations, with a scheme that we introduce here and apply to quenches of the transverse-field Ising model to quantum-critical points in one and two dimensions. The QMC method is generic and can be applied to a wide range of models and non-equilibrium setups.

We discuss the application of the adiabatic perturbation theory to analyze the dynamics in various systems in the limit of slow parametric changes of the Hamiltonian. We first consider a two-level system and give an elementary derivation of the asymptotics of the transition probability when the tuning parameter slowly changes in the finite range. Then we apply this perturbation theory to many-particle systems with low energy spectrum characterized by quasiparticle excitations. Within this approach we derive the scaling of various quantities such as the density of generated defects, entropy and energy. We discuss the applications of this approach to a specific situation where the system crosses a quantum critical point. We also show the connection between adiabatic and sudden quenches near a quantum phase transitions and discuss the effects of quasiparticle statistics on slow and sudden quenches at finite temperatures.

We engineer a quantum bath that enables entropy and energy exchange with a one-dimensional Bose-Hubbard lattice with attractive on-site interactions. We implement this in an array of three superconducting transmon qubits coupled to a single cavity mode; the transmons represent lattice sites and their excitation quanta embody bosonic particles. Our cooling protocol preserves particle number-realizing a canonical ensemble-and also affords the efficient preparation of dark states which, due to symmetry, cannot be prepared via coherent drives on the cavity. Furthermore, by applying continuous microwave radiation, we also realize autonomous feedback to indefinitely stabilize particular eigenstates of the array.Ordinarily, uncontrolled dissipation destroys quantum coherence, but it is now appreciated that an engineered quantum bath is a valuable resource for quantum computation (state preparation [1-4] and system reset [5]) and quantum error correction [6]. A dynamical bath can induce cooling or heating, and is of great utility in optomechanical ground state preparation [7,8], quantum state transfer between various EM modes [9-11], amplification [12-14], many-particle quantum simulation [15,16], and entanglement generation [17,18].Initially conceived and implemented in trapped ion systems [19,20], dissipation engineering has been implemented in solid state systems with two recent experiments on superconducting qubits: autonomously controlling the orientation on the Bloch sphere of a single qubit [3], and stabilizing a Bell-state in a two-qubit system [4]. In this Letter, we experimentally demonstrate that dissipation engineering can be used to control a novel, complex superconducting system embodying a much larger Hilbert space: the ten lowest-lying energy levels of a coupled, three-transmon array which realizes a one-dimensional, attractive Bose-Hubbard Hamiltonian. The techniques employed in this work define a path for cooling and stabilizing complex quantum systems to specific target states.The Bose-Hubbard Hamiltonian [21] is a prototypical model used to describe a broad class of quantum matter. While the repulsive side of the Bose-Hubbard phase diagram has been extensively explored in groundbreaking experiments with ultracold atoms in optical lattices [22][23][24], the attractive regime has thus far eluded emulation. Our realization of this model here, in perhaps its simplest incarnation, opens the door to experimental verification of as-yet unexplored predictions of attractive Bose-Hubbard dynamics: the existence of selfbound states [25][26][27] and the possibility to create largescale multipartite entanglement [28].The cooling protocol we develop here, based on Raman scattering processes, facilitates entropy and energy exchange between the qubit array and its bath while

We discuss two complimentary problems: adiabatic loading of one-dimensional bosons into an optical lattice and merging two one-dimensional Bose systems. Both problems can be mapped to the sine-Gordon model. This mapping allows us to find power-law scalings for the number of excitations with the ramping rate in the regime where the conventional linear response approach fails. We show that the exponent of this power law is sensitive to the interaction strength. In particular, the response is larger, or less adiabatic, for strongly (weakly) interacting bosons for the loading (merging) problem. Our results illustrate that in general the nonlinear response to slow relevant perturbations can be a powerful tool for characterizing properties of interacting systems.Dynamics of one-dimensional (1D) cold gases [1,2,3] is characterized by a strong response to external perturbations [4,5] compared to higher-dimensional systems, due to the large density of low-energy states. For example, for slow processes the excitations are strongly enhanced and can qualitatively affect the dynamics and lead to the breakdown of adiabatic approximation [6]. Understanding slow dynamics has two-fold implications. First, such dynamics can be used as a probe of interacting systems in the regimes where the linear response (Kubo-type) analysis fails. Second, this understanding can help to devise optimal ways for minimizing heating in the system during slow processes. This minimization can be important for various applications including adiabatic preparation of strongly correlated systems and adiabatic quantum computing. In this Letter we will demonstrate both such implications using two related problems of loading 1D interacting bosons into an optical lattice and merging (or splitting) two 1D condensates.The physical parameter that defines the properties of 1D bosonic systems is the ratio of the interaction and kinetic energies γ = mc/( 2 ρ), where m is the mass of a particle, c is the interaction strength [7], and ρ is the density of particles. In the spatially uniform system, the energy spectrum is gapless in the whole range of the parameter γ, which at low energies is reflected in the Luttinger liquid description [8]. In the extreme limit of strong interactions, γ ≫ 1, Tonks-Girardeau (TG) gas [9], the particles behave as hard-core spheres, with the repulsion playing a role similar to the Pauli exclusion principle. This regime has been recently realized in cold atoms [10,11].External perturbations can significantly modify the phase structure of bosonic gases.A bosonic system undergoes the transition from superfluid to Mottinsulator [12,13], driven by the competition of the kinetic and interaction energy: in deep lattices the insulating phase is realized when the interaction energy becomes larger than the tunneling (kinetic) energy. This mean-field picture of the transition is modified in one dimension. If the repulsion between particles is strong enough, an arbitrarily small periodic potential com- mensurate with the boson density stabilizes th...