Abstract This chapter is dedicated to open quantum systems. We explain the origin of non-unitary time evolution by studying different interactions between a qubit and the environment. The first part of the chapter introduces the time evolution equation of an open quantum system, also called the Born-Markov master equation and the Redfield equation. We then study the interaction between a qubit and a thermal bath of harmonic oscillators and introduce the qubit relaxation and the dephasing times. After discussing the relationship between relaxation and dephasing times of a qubit and the power spectral density of the noise source, we present the Hahn echo and the Carr-Purcell-Meiboom-Gill experiments, two protocols that provide information about the noise spectrum. The chapter ends with a discussion of the driven evolution of a qubit subject to spontaneous emission and dephasing
Abstract This chapter is dedicated to open quantum systems. We explain the origin of non-unitary time evolution by studying different interactions between a qubit and the environment. The first part of the chapter introduces the time evolution equation of an open quantum system, also called the Born-Markov master equation and the Redfield equation. We then study the interaction between a qubit and a thermal bath of harmonic oscillators and introduce the qubit relaxation and the dephasing times. After discussing the relationship between relaxation and dephasing times of a qubit and the power spectral density of the noise source, we present the Hahn echo and the Carr-Purcell-Meiboom-Gill experiments, two protocols that provide information about the noise spectrum. The chapter ends with a discussion of the driven evolution of a qubit subject to spontaneous emission and dephasing
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system (S∪E) is isolated from the rest of the universe. So we have two independent Hamiltonians H^S, H^E and an interaction V^ between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction V^ with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time t0=0, with a Gaussian kernel, for 0≤t<tc where tc∈]0,∞] is a critical time such that reversibility is lost for t>tc if tc<+∞. Some manifestations of decoherence, for t>0 small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.
Typically, we are interested in a small system (a few particles, or some region in space), entangled with its surroundings. Exact equations (Dyson or Nakajima–Zwanzig) for the reduced system dynamics are readily derived using suitable projection operators. They are rarely solvable, however, since full account is taken for mutual influences between the system and its environment. Nevertheless, for weak mutual coupling, environment-induced dynamics in the (small) system can be much slower than system-induced dynamics in the (large) environment. This justifies the Born–Markov approximation, leading to closed equations for the system. In Hilbert space, we demonstrate the emergence of irreversible dynamics and exponential decay for pure sates. In Liouville space, we derive the Redfield equation. Invoking the secular (or, rotating-wave) approximation, we derive Pauli’s master equations, which properly account for relaxation to equilibrium. Rates of spontaneous emission, coherence transfer (Bloch equations), and pure dephasing are derived and analyzed for a dissipative qubit.
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.
The act of measurement on a quantum state is supposed to "dephase" (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as "decoherence"), then "decohere" and "collapse" (or more precisely "register" and "reduce") the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger's equation. Progress has, however, been made in studying this problem with two main calculation tools-one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the "consistent" or "decoherent"' histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the "environmental decoherence" approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the "environmental decoherence" approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with "dephasing" followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly "un-collapse" if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this "dephasing" quickly leads to "decoherence"-collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner's friend experiment, as well as the Mott experiment, which is explored in "Connection to some general theoretical results" and "Recurrence times" (especially the enumerated points in "Recurrence times"). The upshot is that as long as Wigner's friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner's friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper-we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.