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arXiv:1303.5831 · Physical Review B

Transient dynamics of open quantum systems

Oleksiy Kashuba, Herbert Schoeller

Published May 22, 2013• 19 citations
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Abstract

We present a renormalization-group (RG) method which allows for an analytical study of the transient dynamics of open quantum systems on all time scales. Whereas oscillation frequencies and decay rates of exponential time evolution follow from the fixed point positions, the long-time behavior of preexponential functions is related to the scaling behavior around the fixed points. We show that certain terms of the RG flow are only cut off by inverse time, which leads to a difference between infrared and ultraviolet scaling. An evaluation for the Ohmic spin boson model at weak damping reveals significant deviations from previous predictions in the long-time regime. We propose that weak-coupling problems for stationary quantities can in principle turn into strong-coupling ones for the determination of the long-time behavior.

DOI
10.1103/physrevb.87.201402

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We present a renormalization-group (RG) method which allows for an analytical study of the transient dynamics of open quantum systems on all time scales. Whereas oscillation frequencies and decay rates of exponential time evolution follow from the fixed point positions, the long-time behavior of preexponential functions is related to the scaling behavior around the fixed points. We show that certain terms of the RG flow are only cut off by inverse time, which leads to a difference between infrared and ultraviolet scaling. An evaluation for the Ohmic spin boson model at weak damping reveals significant deviations from previous predictions in the long-time regime. We propose that weak-coupling problems for stationary quantities can in principle turn into strong-coupling ones for the determination of the long-time behavior.
A tutorial introduction is presented for the calculation of the time dynamics for models of dissipative quantum mechanics where a small quantum system is coupled to noninteracting bosonic or fermionic reservoirs. We discuss the basics and applications of the real-time renormalization group method (RTRG), where the Fourier variable EEE conjugate to the time variable ttt is used as flow parameter. It is shown how a well-controlled perturbation theory can be set up in the coupling to the bath where secular and logarithmic terms are resummed to all order of perturbation theory. This is achieved by a set of perturbative RG equations for the effective Liouvillian and the effective vertices. Truncating the RG equations we show how the time dynamics can be obtained analytically for exponentially small, intermediate and exponentially large times. Basic applications are discussed for particle and potential energy exchange within the interacting resonant level model, spin exchange for the nonequilibrium Kondo model, and heat exchange within the ohmic spin boson model at zero bias.
We present the T-flow renormalization group method, which computes the memory kernel for the density-operator evolution of an open quantum system by lowering the physical temperature T of its environment. This has the key advantage that it can be formulated directly in real time, making it particularly suitable for transient dynamics, while automatically accumulating the full temperature dependence of transport quantities. We solve the T-flow equations numerically for the example of the single impurity Anderson model. We benchmark in the stationary limit, readily accessible in real-time for voltages on the order of the coupling or larger using results obtained by the functional renormalization group, density-matrix renormalization group and the quantum Monte Carlo method. Here we find quantitative agreement even in the worst case of strong interactions and low temperatures, indicating the reliability of the method. For transient charge currents we find good agreement with results obtained by the 2PI Green's function approach. Furthermore, we analytically show that the short-time dynamics of both local and non-local observables follow a "universal" temperature-independent behaviour when the metallic reservoirs have a flat wide band.
We apply the recently developed real-time renormalization-group (RTRG) method to different quantum dissipative systems. In contrast to other renormalization-group techniques the RTRG approach is formulated on the Keldysh contour and therefore allows for the description of nonequilibrium situations. The method is based on an exact kinetic equation which determines the time-dependent reduced density matrix. The kinetic equation involves an integral kernel which describes the phenomenon of dissipation. We calculate this quantity using the RTRG method: After applying Wick's theorem the arising diagrams are accounted for successively with respect to their time scale. During that process the theory is left invariant by renormalizing the operators describing the theory. As a starting point we discuss the polaron problem. We calculate the renormalized ground-state energy and the effective mass. We compare our results of the RTRG with those obtained from the flow equation formalism, which is based on infinitesimal transformations. In the central part of this thesis we consider the spin-boson model which is applicable to various physical and chemical systems. It models a particle in a double-well potential which is coupled to a bosonic environment. We determine static properties such as the static susceptibility. We also present results for the time evolution of the reduced density matrix starting from an arbitrary nonequilibrium state. The asymptotic behaviour of the reduced density matrix can be characterized by an oscillation frequency and two decay constants, all of which are determined. Finally, we calculate equilibrium correlation functions. We find consistency of our results with quantum Monte Carlo simulations and with the Shiba-relation. Additionally, our results show the correct scaling behaviour. Comparisons are also drawn to results of the noninteracting blip approximation and to results obtained for the anisotropic Kondo model. Our approach is very accurate for not too large couplings to the environment. Furthermore, the stationary tunnel current through a double quantum dot coupled to a phonon bath is calculated and compared to experimental data. Eventually, the running couplings of the two-lead Kondo model are discussed.
We develop a semianalytical approach beyond the Born-Markov approximation to study the quench dynamics of the spin-boson model in the strong-coupling regime ($\ensuremath{\alpha}\ensuremath{\le}1/2$) for the Ohmic bath. The basic idea in our approach is to write an effective time-dependent model for the dynamics of the system coupled to the bosonic bath after integrating out high-frequency bath modes. By applying this procedure to the Heisenberg equations of motion, we derive a set of flow equations for the system parameters as a function of time. The final flow equations look similar to those of the equilibrium renormalization-group theory; however, in our derivation the scaling parameter is set by the real time. We solve the equations of motion with time-dependent renormalized parameters and show that the resulting dynamics is in decent agreement with the exact NRG calculations as well as the noninteracting blip approximation that is a well-known good solution in this limit.
Open system probes can provide an efficient means to characterize quantum many-body systems by employing them as engineered environments. The key idea is to map long-range spatial correlations of the environment onto dynamical correlations in the evolution of a simple quantum probe. Using the example of a qubit coupled to a transverse-field Ising model, we show how the non-Markovian rate or spectral flow can be used to identify stable and unstable fixed points, infer scaling dimensions of relevant fields, and deduce the renormalization group flow induced by deformations around any fixed point.
We study quantum many-body effects on the real-time evolution of the current through quantum dots. By using a non-equilibrium renormalization group approach, we provide analytic results for the relaxation dynamics into the stationary state and identify the microscopic cutoff scales that determine the transport rates. We find rich non-equilibrium physics induced by the interplay of the different energy scales. While the short-time limit is governed by universal dynamics, the long-time behavior features characteristic oscillations as well as an interplay of exponential and power-law decay.