Gotowa bibliografia na temat „Quantum trajectory framework”

One of the important and famous topics in general theory of relativity and gravitation is the problem of geodesic deviation and its related singularity theorems. An interesting subject is the investigation of these concepts when quantum effects are considered. Since the definition of trajectory is not possible in the framework of standard quantum mechanics (SQM), we investigate the problem of geodesic equation and its related topics in the framework of Bohmian quantum mechanics in which the definition of trajectory is possible. We do this in a fixed background and we do not consider the backreaction effects of matter on the space–time metric.

Particle tunneling from a quantum corrected black hole in the gravity's rainbow was investigated by the radial trajectory method of the tunneling framework. Using the thermodynamic property of the event horizon, a simpler method for calculating the tunneling probability was shown. In this method, the Painleve coordinate transformation of spacetime and the radial trajectory equation of the tunneling particles used in the previous radial trajectory method was not used. Using the simpler method, the tunneling probability of outgoing particles, regardless of whether they are massless or massive, were calculated in a unified way. The emission rates were related to the changes of the black hole entropies before and after the emission. This implies that the emission spectrum agrees with the underling unitary theory. In addition, the Bekenstein–Hawking area for the modified black hole was established and the emission spectrum with quantum corrections was discussed.

Quantum trajectory theory is the best mathematical set up to model continual observations of a quantum system and feedback based on the observed output. Inside this framework, we study how to enhance the squeezing of the fluorescence light emitted by a two-level atom, stimulated by a coherent monochromatic laser. In the presence of a Wiseman-Milburn feedback scheme, based on the homodyne detection of a fraction of the emitted light, we analyze the squeezing dependence on the various control parameters.

We investigate a composite quantum collision model with measurements on the memory part, which effectively probe the system. The framework allows us to adjust the measurement strength, thereby tuning the dynamical map of the system. For a two-qubit setup with a symmetric and informationally complete measurement on the memory, we study the divisibility of the resulting dynamics in dependence of the measurement strength. The measurements give rise to quantum trajectories of the system and we show that the average asymptotic purity depends on the specific form of the measurement. With the help of numerical simulations, we demonstrate that the different performance of the measurements is generic and holds for almost all interaction gates between the system and the memory in the composite collision model. The discrete model is then extended to a time-continuous limit.

We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler–Lagrange equations is obtained from a higher-order Hamilton–Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre–Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton–Pontryagin principle and preserves the geometric properties of the continuous-time solution.

Quantum cosmology based on the Wheeler De Witt equation represents a simple way to implement plausible quantum effects in a gravitational setup. In its minisuperspace version wherein one restricts attention to FLRW metrics with a single scale factor and only a few degrees of freedom describing matter, one can obtain exact solutions and thus acquire full knowledge of the wave function. Although this is the usual way to treat a quantum mechanical system, it turns out however to be essentially meaningless in a cosmological framework. Turning to a trajectory approach then provides an effective means of deriving physical consequences.

Shannon's theory of information was built on the assumption that the information carriers were classical systems. Its quantum counterpart, quantum Shannon theory, explores the new possibilities arising when the information carriers are quantum systems. Traditionally, quantum Shannon theory has focused on scenarios where the internal state of the information carriers is quantum, while their trajectory is classical. Here we propose a second level of quantization where both the information and its propagation in space–time is treated quantum mechanically. The framework is illustrated with a number of examples, showcasing some of the counterintuitive phenomena taking place when information travels simultaneously through multiple transmission lines.

A universal scheme is introduced to speed up the dynamics of a driven open quantum system along a prescribed trajectory of interest. This framework generalizes counterdiabatic driving to open quantum processes. Shortcuts to adiabaticity designed in this fashion can be implemented in two alternative physical scenarios: one characterized by the presence of balanced gain and loss, the other involves non-Markovian dynamics with time-dependent Lindblad operators. As an illustration, we engineer superadiabatic cooling, heating, and isothermal strokes for a two-level system, and provide a protocol for the fast thermalization of a quantum oscillator.

Thesis (Ph. D.)--University of Oregon, 2003.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 129-135). Also available for download via the World Wide Web; free to University of Oregon users.

Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement. The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the error rate is less than the rate at which weak measurement extracts information. It is also obvious that error correction with too small a measurement strength should be avoided.

Abstract After an introduction to the phenomenological theory of non-equilibrium thermodynamics, this theory is derived and extended forsmall systems described by a classical Markov process obeying local detailed balance. Thermodynamic definitions for internal energy, heat, work, entropy and entropy production are provided along a single stochastic trajectory. It is shown that the fluctuations in work and entropy production satisfy universal constraints, known as fluctuation theorems. By providing an independent derivation of them starting from microscopically reversible Hamiltonian dynamics in the full system-bath phase space, it is demonstrated that fluctuation theorems also hold in the non-Markovian regime. The theoretical framework established here is called (classical) stochastic thermodynamics. It has found widespread applications in biology and biochemistry, soft condensed matter physics as well as various artificial nanostructures down to the quantum regime. The chapter finishes with a discussion of the particularly relevant setting of single-molecule pulling experiments.

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.

The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.