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arXiv:1802.10014

Dynamics of open quantum systems

Herbert Schoeller

Published Feb 27, 2018• 10 citations
arXivHTMLPDF

Abstract

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 $E$ conjugate to the time variable $t$ 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.

DOI
10.48550/arxiv.1802.10014

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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 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 present the real-time renormalization group (RTRG) method as a method to describe the stationary state current through generic multilevel quantum dots in nonequilibrium. The employed approach consists of a very rudimentary approximation for the renormalization group (RG) equations which neglects all vertex corrections while it provides a means to compute the effective dot Liouvillian self-consistently. Being based on a weak-coupling expansion in the tunneling between dot and reservoirs, the RTRG approach turns out to reliably describe charge fluctuations in and out of equilibrium for arbitrary coupling strength, even at zero temperature. We confirm this in the linear response regime with a benchmark against highly accurate numerical renormalization group data in the exemplary case of three-level quantum dots. For small to intermediate bias voltages and weak Coulomb interactions, we find an excellent agreement between RTRG and functional renormalization group data, which can be expected to be accurate in this regime. As a consequence, we advertise the presented RTRG approach as an efficient and versatile tool to describe charge fluctuations in quantum dot systems.
We introduce a real time version of the functional renormalization group, which allows us to study correlation effects on nonequilibrium transport through quantum dots. Our method is equally capable to address (i) the relaxation out of a nonequilibrium initial state into a (potentially) steady state driven by a bias voltage and (ii) the dynamics governed by an explicitly time-dependent Hamiltonian. All time regimes from transient to asymptotic can be tackled; the only approximation is the consistent truncation of the flow equations at a given order. As an application we investigate the relaxation dynamics of the interacting resonant level model, which describes a fermionic quantum dot dominated by charge fluctuations. Moreover, we study incoherence and relaxation phenomena within the ohmic spin-boson model by mapping the latter to the interacting resonant level model.
We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. We compare this dynamics with the exact dynamics in an exactly solvable spin-boson model.
Using a real-time renormalization-group method we determine the complete dynamics of the spin-boson model with Ohmic dissipation for coupling strengths $\ensuremath{\alpha}\ensuremath{\lesssim}0.1--0.2.$ We calculate the relaxation and dephasing time, static susceptibility, and correlation functions. Our results are consistent with quantum Monte Carlo simulations and the Shiba relation. We present reliable results for finite cutoff and finite bias in a regime where perturbation theory in $\ensuremath{\alpha}$ or in tunneling breaks down. Furthermore, an unambiguous comparison to results from the Kondo model is achieved.
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.