Transport theory yields renormalization group equations

We show that dissipative transport and renormalization can be described in a single theoretical framework. The appropriate mathematical tool is the Nakajima-Zwanzig projection technique. We illustrate our result in the case of interacting quantum gases, where we use the Nakajima-Zwanzig approach to investigate the renormalization group flow of the effective two-body interaction.Comment: 11 pages REVTeX, twocolumn, no figures; revised version with additional examples, to appear in Phys. Rev.

Lagrange formalism of memory circuit elements: classical and quantum formulations

The general Lagrange-Euler formalism for the three memory circuit elements, namely, memristive, memcapacitive, and meminductive systems, is introduced. In addition, {\it mutual meminductance}, i.e. mutual inductance with a state depending on the past evolution of the system, is defined. The Lagrange-Euler formalism for a general circuit network, the related work-energy theorem, and the generalized Joule's first law are also obtained. Examples of this formalism applied to specific circuits are provided, and the corresponding Hamiltonian and its quantization for the case of non-dissipative elements are discussed. The notion of {\it memory quanta}, the quantum excitations of the memory degrees of freedom, is presented. Specific examples are used to show that the coupling between these quanta and the well-known charge quanta can lead to a splitting of degenerate levels and to other experimentally observable quantum effects

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

Decoherent time-dependent transport beyond the Landauer-B\"uttiker formulation: a quantum-drift alternative to quantum jumps

We present a model for decoherence in time-dependent transport. It boils down into a form of wave function that undergoes a smooth stochastic drift of the phase in a local basis, the Quantum Drift (QD) model. This drift is nothing else but a local energy fluctuation. Unlike Quantum Jumps (QJ) models, no jumps are present in the density as the evolution is unitary. As a first application, we address the transport through a resonant state $\left\vert 0\right\rangle$ that undergoes decoherence. We show the equivalence with the decoherent steady state transport in presence of a B\"{u}ttiker's voltage probe. In order to test the dynamics, we consider two many-spin systems whith a local energy fluctuation. A two-spin system is reduced to a two level system (TLS) that oscillates among $\left\vert 0\right\rangle$ $\equiv$ $\left\vert \uparrow \downarrow \right\rangle$ and $\left\vert 1\right\rangle \equiv$ $\left\vert \downarrow \uparrow \right\rangle$. We show that QD model recovers not only the exponential damping of the oscillations in the low perturbation regime, but also the non-trivial bifurcation of the damping rates at a critical point, i.e. the quantum dynamical phase transition. We also address the spin-wave like dynamics of local polarization in a spin chain. The QD average solution has about half the dispersion respect to the mean dynamics than QJ. By evaluating the Loschmidt Echo (LE), we find that the pure states $\left\vert 0\right\rangle$ and $\left\vert 1\right \rangle$ are quite robust against the local decoherence. In contrast, the LE, and hence coherence, decays faster when the system is in a superposition state. Because its simple implementation, the method is well suited to assess decoherent transport problems as well as to include decoherence in both one-body and many-body dynamics.Comment: 10 pages, 5 figure

Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics

A new method of deriving reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a set of resolved variables that define a model reduction, the quasi-equilibrium ensembles associated with the resolved variables are employed as a family of trial probability densities on phase space. The residual that results from submitting these trial densities to the Liouville equation is quantified by an ensemble-averaged cost function related to the information loss rate of the reduction. From an initial nonequilibrium state, the statistical state of the system at any later time is estimated by minimizing the time integral of the cost function over paths of trial densities. Statistical closure of the underresolved dynamics is obtained at the level of the value function, which equals the optimal cost of reduction with respect to the resolved variables, and the evolution of the estimated statistical state is deduced from the Hamilton-Jacobi equation satisfied by the value function. In the near-equilibrium regime, or under a local quadratic approximation in the far-from-equilibrium regime, this best-fit closure is governed by a differential equation for the estimated state vector coupled to a Riccati differential equation for the Hessian matrix of the value function. Since memory effects are not explicitly included in the trial densities, a single adjustable parameter is introduced into the cost function to capture a time-scale ratio between resolved and unresolved motions. Apart from this parameter, the closed equations for the resolved variables are completely determined by the underlying deterministic dynamics