Comparing different non-Markovianity measures: A case study

We consider two recently proposed measures of non-Markovianity applied to a particular quantum process describing the dynamics of a driven qubit in a structured reservoir. The motivation of this study is twofold: on one hand, we study the differences and analogies of the non-Markovianity measures and on the other hand, we investigate the effect of the driving force on the dissipative dynamics of the qubit. In particular we ask if the drive introduces new channels for energy and/or information transfer between the system and the environment, or amplifies existing ones. We show under which conditions the presence of the drive slows down the inevitable loss of quantum properties of the qubit.Comment: 5 pages, no figures. Published version with minor modification

Markovian evolution of strongly coupled harmonic oscillators

We investigate how to model Markovian evolution of coupled harmonic oscillators, each of them interacting with a local environment. When the coupling between the oscillators is weak, dissipation may be modeled using local Lindblad terms for each of the oscillators in the master equation, as is commonly done. When the coupling between oscillators is strong, this model may become invalid. We derive a master equation for two coupled harmonic oscillators which are subject to individual heat baths modeled by a collection of harmonic oscillators, and show that this master equation in general contains non-local Lindblad terms. We compare the resulting time evolution with that obtained for dissipation through local Lindblad terms for each individual oscillator, and show that the evolution is different in the two cases. In particular, the two descriptions give different predictions for the steady state and for the entanglement between strongly coupled oscillators. This shows that when describing strongly coupled harmonic oscillators, one must take great care in how dissipation is modeled, and that a description using local Lindblad terms may fail. This may be particularly relevant when attempting to generate entangled states of strongly coupled quantum systems.Comment: 11 pages, 4 figures, significantly revised and close to the published versio

Geometric phase for an adiabatically evolving open quantum system

We derive an elegant solution for a two-level system evolving adiabatically under the influence of a driving field with a time-dependent phase, which includes open system effects such as dephasing and spontaneous emission. This solution, which is obtained by working in the representation corresponding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic and geometric phases of the evolving density matrix to be separated and relatively easily calculated.Comment: 10 pages, 0 figure

Canonical form of master equations and characterization of non-Markovianity

Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.Comment: v2: Significant update, with many new results and one new author. 12 pages; v3: Minor clarifications, to appear in PRA; v4: matches published versio

Quantum probability rule : a generalization of the theorems of Gleason and Busch

Buschs theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleasons theorem. Here we show that a further generalization is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction-retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory

Revisiting the damped quantum harmonic oscillator

We reanalyse the quantum damped harmonic oscillator, introducing three less than common features. These are (i) the use of a continuum model of the reservoir rather than an ensemble of discrete oscillators, (ii) an exact diagonalisation of the Hamiltonian by adapting a technique pioneered by Fano, and (iii) the use of the thermofield technique for describing a finite temperature reservoir. We recover in this way a number of well-known and some, perhaps, less familiar results. An example of the latter is an ab initio proof that the oscillator relaxes to the mean-force Gibbs state. We find that special care is necessary when comparing the damped oscillator with its undamped counterpart as the former has two distinct natural frequencies, one associated with short time evolution and the other with longer times

Quantum-classical correspondence in spin-boson equilibrium states at arbitrary coupling

It is known that the equilibrium properties of nanoscale systems can deviate significantly from standard thermodynamics due to their coupling to an environment. For the generalized $\theta$-angled spin-boson model, here we derive an explicit form of the classical mean force equilibrium state. Taking the large spin limit of the quantum spin-boson model, we demonstrate that the quantum-classical correspondence is maintained at arbitrary coupling strength. This correspondence gives insight into the conditions for a quantum system to be well-approximated by its classical counterpart. We further demonstrate that, counterintuitively, previously identified environment-induced 'coherences' in the equilibrium state of weakly coupled quantum spins, do not disappear in the classical case. Finally, we categorise various coupling regimes, from ultra-weak to ultra-strong, and find that the same value of coupling strength can either be 'weak' or 'strong', depending on whether the system is quantum or classical. Our results shed light on the interplay of quantum and mean force corrections in equilibrium states of the spin-boson model, and will help draw the quantum to classical boundary in a range of fields, such as magnetism and exciton dynamics

Completely positive maps with memory

The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. This paper examines master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel. A physical model that possesses such an exponential memory kernel is presented. This model contains a classical, fluctuating environment based on random telegraph signal stochastic variables.Comment: 4 page

Measurement master equation

We derive a master equation describing the evolution of a quantum system subjected to a sequence of observations. These measurements occur randomly at a given rate and can be of a very general form. As an example, we analyse the effects of these measurements on the evolution of a two-level atom driven by an electromagnetic field. For the associated quantum trajectories we find Rabi oscillations, Zeno-effect type behaviour and random telegraph evolution spawned by mini quantum jumps as we change the rates and strengths of measurement.Comment: 14 pages and 8 figures, Optics Communications in pres

A quantum trajectory analysis of Berry's Phase in a dissipative environment

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