This chapter explores to what extent the closed-contour Schwinger–Keldysh approach and the Lindblad Master equation can be connected with each other. Here, the connection with the Schwinger–Keldysh closed-contour approach does not involve the full machinery of the Green’s functions method, but rather refers directly to the time evolution of the many-body density matrix, which contains a forward evolution operator from the reference time t ₀ to the measuring time t and a backward evolution operator from t back to t ₀. The key approximations to derive the Lindblad Master equation are specified in detail. As an example, a two-level system coupled to a phonon bath is explicitly considered.
This chapter explores to what extent the closed-contour Schwinger–Keldysh approach and the Lindblad Master equation can be connected with each other. Here, the connection with the Schwinger–Keldysh closed-contour approach does not involve the full machinery of the Green’s functions method, but rather refers directly to the time evolution of the many-body density matrix, which contains a forward evolution operator from the reference time t ₀ to the measuring time t and a backward evolution operator from t back to t ₀. The key approximations to derive the Lindblad Master equation are specified in detail. As an example, a two-level system coupled to a phonon bath is explicitly considered.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
This chapter introduces the contour Schwinger–Keldysh method for time-dependent averages, in light of its relevance to nonequilibrium processes. A key feature of this approach is that it leaves open the possibility that no state of a system in the future can be identified with any of its states in the past. This method is here illustrated in detail with reference to time-dependent quantum averages, whereby for definiteness the system is initially prepared at the reference time t ₀ in a definite quantum state.
This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.
We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.
The contour-ordered Green's function considered in the previous chapter was ideal for discussing general closed time path properties such as the perturbative diagrammatic structure for non-equilibrium states. However, the contour-ordered Green's function lacks physical transparency and does not appeal to intuition. We need a different approach, which brings quantities back to real time. To accomplish this we introduce a representation where forward and return parts of the closed time path are ordered by numbers, specifying the position of a contour time by two indices, i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique then formulated in a fashion where the aspects of non-equilibrium states emerge in the physically most appealing way. In particular, we shall construct the representation where spectral properties and quantum statistics show up on a different footing in the diagrams. Lastly, we consider the connection to the imaginary-time treatment of non-equilibrium states, and establish its equivalence to the real-time approach propounded in this chapter. Real-time matrix representation To let our physical intuition come into play; we need to get from contour times back to real times. This is achieved by labeling the forward and return contours of the closed time path, depicted in Figure 4.5, by numbers, specifying the position of a contour time by an index. The forward contour we therefore label c 1 and the return contour c 2 , i.e. a contour time variable gets tagged by the label 1 or 2 specifying its belonging to forward or return contour, respectively.
This chapter considers the product of operators in the Heisenberg representation and express it in terms of the contour time-ordering operator. Since the relative order in which the operators enter the quantum average matters, this order has to be specified in detail. This procedure leads to considering the single- and two-particle Green’s functions, where the product contains, respectively, two and four field operators, which are at the core of the diagrammatic many-particle theory to be developed in what follows.