We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.
We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.
Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short-distance and long-distance degrees of freedom (d.o.f.), coupled via the Hamiltonian. Observations using purely long-distance observables are described by the reduced density matrix that arises from tracing out the short-distance d.o.f. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}>\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}$, in which the coupling between high- and low-energy d.o.f. is treated to second order in perturbation theory. We then describe the equations of motion under suitable time averaging, reflecting the limited time resolution of actual experiments, and find an expansion of the master equation in powers of $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}/\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}$, after the fashion of effective field theory. The failure of the system to be Hamiltonian or even Markovian appears at higher orders in this ratio. We compute the evolution of the density matrix in three specific examples: coupled spins, linearly coupled simple harmonic oscillators, and an interacting scalar quantum field theory. Finally, we argue that the logarithm of the Feynman-Vernon influence functional is the correct analog of the Wilsonian effective action for this problem.
We compare different quantum master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad-type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse-graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.
We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.
Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relies on effective Hamiltonian models that introduce additional nonlinear processes which approximate the long-time dynamics so that highly oscillatory terms may be ignored. However, the removal of such high frequency transitions can only be performed rigorously within a systematic framework of time-coarse graining, which is a fundamentally irreversible operation. This implies that standard approaches with unitary effective models cannot accurately capture the long-time behavior of strongly driven nonlinear quantum systems in general. We introduce a systematic perturbation theory for obtaining the complete non-unitary effective model of the time-coarse grained (TCG) dynamics of a driven quantum system to any order in the coupling strengths. We derive a closed-form analytical formula for both unitary and non-unitary contributions, in the form of an effective Hamiltonian and non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory presumes unitary time evolution at the microscopic level, the time-coarse grained dynamics is found to follow a non-unitary time evolution in general. This occurs even when there is no open heat reservoir for the system to become entangled with or dissipate into. We demonstrate the effectiveness of the new method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.
Real-world experiments on quantum systems are always performed with measurement apparatus whose interaction times with the systems are finite. This restricts the observable quantum states to the space of time-coarse-grained density matrices ${\overline{\ensuremath{\rho}}(t)}$, providing motivation for the time-coarse-graining (TCG) approach that solves for $\overline{\ensuremath{\rho}}(t)$ without ever referring to the unobservable $\ensuremath{\rho}(t)$ of infinite time resolution. Phenomenologically, this implies that coherent transitions far outside the bandwidth would be filtered out, leaving only their effective impacts on the ``slow'' dynamics resolvable by the finite time resolution of the measurements. Therefore, the TCG framework provides rigorous justification for many existing effective Hamiltonian methods in the literature that aim at capturing the unitary part of the long-time dynamics. However, since time-coarse-graining is fundamentally irreversible, the resulting effective model allows for secular loss of information and dissipation of energy in general, which cannot be captured by any unitary effective models and has to be treated with explicit time-coarse-graining. Such incoherent effects are particularly prominent in driven nonlinear quantum systems where exchange of information and energy with the drive gives rise to incoherent effective dynamics at all finite time resolutions in general. While rigorous in principle, existing TCG methods in the literature can be applied only to simple systems by one iteratively solving superoperator equations at low orders in the coupling strengths. The complexity of such methods prevents systematic study of the time-coarse-grained dynamics and limits analytical results to the IR (low-resolution) limit in most cases. We address these limitations in this paper, presenting a systematic time-coarse-graining method that overcomes the complexities and restrictions of current techniques, offering a comprehensive and accurate modeling framework for driven nonlinear quantum systems. We derive closed-form formulas as well as diagrammatic representations for both unitary and nonunitary contributions, in the form of an effective Hamiltonian and nonunitary dissipators at arbitrary order in the coupling strengths, and complement this with a computer-algebra software package. We demonstrate the effectiveness of the method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.
A generating functional formalism is developed to facilitate the derivation of coarse-grained dynamics of macroscopically relevant variables in various types of many-body problems. The relevant variables can be found either by using symmetry analysis of the action or experimental data. The outcome of the coarse-graining process is a set of coupled partial differential equations for the expectation values of the relevant observables. The set of the macroscopically important observables consists typically of conserved variables and order operators, which are related to the broken symmetries of the system. Symmetry breaking (spontaneous or explicit) in the formalism can be dynamically induced by coupling the system to a generalized heat bath.\n\nThe nonequilibrium generating functional formalism is applied to coarse-graining of generalized hydrodynamics of macroscopic quantum phenomena such as Bose fluids and superconductors, and finally classical fluids. The method is by no means restricted to fluids, but can be applied to condensed matter problems as well. High density systems are in general more problematic and it remains to be seen if one can come up with more powerful approximation methods within the current formalism in the future.\n\nThe generating functional formalism is shown to reproduce in the specific limits the well-known ground state, finite temperature and time-dependent density functional theories in the quantum regime. We enlarge the set of relevant variables to include also non-conserved order-operators and show how to form generalized density functional theories, which can be by construction linked to the phase-transition behaviour of the system. They are extensions of the traditional density functional theories that were not developed to describe phase changes but only facilitated the expression of the system's (free) energy in terms of the conserved density variable (e.g. number density of electrons).\n\nWe develop a generalized density functional theory (effective action of the generating functional) for superconductors and superfluids. For superconductors it is shown by simple symmetry analysis that the relevant order parameter is the pair field of two electrons. Coarse-graining out the phononic degrees of freedom is shown to lead to a retarded effective interaction between the electrons. In the local static limit (effective interaction between electrons static) the theory reduces to the time-dependent Ginzburg-Landau theory. Similarly, by symmetry arguments, the order parameter of a weakly interacting Bose fluid is identified and its dynamics in the tree level approximation is shown to follow the Gross-Pitaevskii equation. Higher order corrections can be systematically produced.\n\nClassical molecular dynamics and stochastic cellular automata types of higher level descriptions can be mapped into the same formalism using commonly known techniques. It is also shown how to connect the static and dynamic density functionals to phase-field models (models of critical dynamics), which are commonly used in materials science modelling. A specific example is provided for a liquid-gas-solid system. The phase-field modelling is then applied to the derivation of dynamics of the liquid-gas phase boundaries and triple lines for which a new projection operator technique is developed. We explain in detail how to go from microscopics to macroscopics and relate the effective parameters on each level to each other. For example, we express the macroscopic surface tensions from the phase-field model in terms of the molecular dynamics simulation parameters. We also discuss how to extract the relevant cross-over time and length scales from the theoretical description and relate them to the experimentally measurable ones. It is shown how the coupling of the triple line and the liquid-gas boundary leads to emergence of memory effects. For confined geometry we also derive a new form for the restoring force acting on the contact line. The phase-field model of the liquid-gas-solid system is first discussed without explicit hydrodynamics from the point of view of diffuse mass transfer and then hydrodynamic effects (momentum conservation) are added.\n\nEvolution equations are derived for the mass, momentum and energy densities representing the proper set of macro variables of simple fluids. The derivation is based on a macroscopic Poisson bracket formulation of phase-field dynamics. Navier-Stokes equations are obtained with extra force terms arising from wall-fluid interactions and capillary stresses. It is also shown how the hydrodynamics boundary conditions for stress tensor can be derived from the bulk fields alone in the sharp interface limit. Some mechanisms for generation of the no-slip boundary condition are also hypothesized. These results are important because they show how in general boundary conditions and corrections to them can be derived and how the numerical implementation can benefit from such a formulation of hydrodynamics, where all boundary conditions are expressed in terms of bulk fields alone. Again, these results have wider applicability than the setting where they have been derived.\n\nThe advantage of a systematic coarse-graining procedure enables us to clarify the role of the order parameter concept in the phase-field description of one and two component fluids. We also consider different ways of including gravitational effects and apply dimensional analysis to obtain a condition for the onset of capillary turbulence.\n\nFinally, we present some new formalism for stochastic processes, where the relevant field variables are embedded in an environment characterized by quenched disorder. Whenever some lower dimensional (e.g. defect) structures are projected out of the bulk field evolution equations containing either frozen of coarse-graining generated noise fields, a more complicated effective stochastic noise component will appear in the projected equation of motion of the defect. In particular, the effective noise can be long-ranged even if the quenched background disorder is not. Several condensed matter and hydrodynamical examples describing dynamics of interfaces and line-like objects such as polymers and domain wall defects are considered.