Finding the Kraus decomposition from a master equation and vice versa

For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is reviewed for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N^2 x N^2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalising a related N^2 x N^2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a `best possible' master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.Comment: 16 pages, no figures. Appeared in special issue for conference QEP-16, Manchester 4-7 Sep 200

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

Single microwave photon detection in the micromaser

High efficiency single photon detection is an interesting problem for many areas of physics, including low temperature measurement, quantum information science and particle physics. For optical photons, there are many examples of devices capable of detecting single photons with high efficiency. However reliable single photon detection of microwaves is very difficult, principally due to their low energy. In this paper we present the theory of a cascade amplifier operating in the microwave regime that has an optimal quantum efficiency of 93%. The device uses a microwave photon to trigger the stimulated emission of a sequence of atoms where the energy transition is readily detectable. A detailed description of the detector's operation and some discussion of the potential limitations of the detector are presented.Comment: 8 pages, 5 figure

Test of the quantumness of atom-atom correlations in a bosonic gas

It is shown how the quantumness of atom-atom correlations in a trapped bosonic gas can be made observable. Application of continuous feedback control of the center of mass of the atomic cloud is shown to generate oscillations of the spatial extension of the cloud, whose amplitude can be directly used as a characterization of atom-atom correlations. Feedback parameters can be chosen such that the violation of a Schwarz inequality for atom-atom correlations can be tested at noise levels much higher than the standard quantum limit

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

The equilibrium properties of nanoscale systems can deviate significantly from standard thermodynamics due to their coupling to an environment. We investigate this here for the θ-angled spin–boson model, where we first derive a compact and general form of the classical equilibrium state including environmental corrections to all orders. Secondly, for the quantum spin–boson model we prove, by carefully taking a large spin limit, that Bohr’s quantum–classical correspondence persists at all coupling strengths. This shows, for the first time, the validity of the quantum–classical correspondence for an open system and gives insight into the regimes where the quantum system is well-approximated by a classical one. Finally, we provide the first classification of the coupling parameter regimes for the spin–boson model, from weak to ultrastrong, both for the quantum case and the classical setting. 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

Jump-like unravelings for non-Markovian open quantum systems

Non-Markovian evolution of an open quantum system can be `unraveled' into pure state trajectories generated by a non-Markovian stochastic (diffusive) Schr\"odinger equation, as introduced by Di\'osi, Gisin, and Strunz. Recently we have shown that such equations can be derived using the modal (hidden variable) interpretation of quantum mechanics. In this paper we generalize this theory to treat jump-like unravelings. To illustrate the jump-like behavior we consider a simple system: A classically driven (at Rabi frequency $\Omega$) two-level atom coupled linearly to a three mode optical bath, with a central frequency equal to the frequency of the atom, $\omega_0$, and the two side bands have frequencies $\omega_0\pm\Omega$. In the large $\Omega$ limit we observed that the jump-like behavior is similar to that observed in this system with a Markovian (broad band) bath. This is expected as in the Markovian limit the fluorescence spectrum for a strongly driven two level atom takes the form of a Mollow triplet. However the length of time for which the Markovian-like behaviour persists depends upon {\em which} jump-like unraveling is used.Comment: 11 pages, 5 figure

Local in time master equations with memory effects: Applicability and interpretation

Non-Markovian local in time master equations give a relatively simple way to describe the dynamics of open quantum systems with memory effects. Despite their simple form, there are still many misunderstandings related to the physical applicability and interpretation of these equations. Here we clarify these issues both in the case of quantum and classical master equations. We further introduce the concept of a classical non-Markov chain signified through negative jump rates in the chain configuration.Comment: Special issue on loss of coherence and memory effects in quantum dynamics, J. Phys. B., to appea

Multiple-time correlation functions for non-Markovian interaction: Beyond the Quantum Regression Theorem

Multiple time correlation functions are found in the dynamical description of different phenomena. They encode and describe the fluctuations of the dynamical variables of a system. In this paper we formulate a theory of non-Markovian multiple-time correlation functions (MTCF) for a wide class of systems. We derive the dynamical equation of the {\it reduced propagator}, an object that evolve state vectors of the system conditioned to the dynamics of its environment, which is not necessarily at the vacuum state at the initial time. Such reduced propagator is the essential piece to obtain multiple-time correlation functions. An average over the different environmental histories of the reduced propagator permits us to obtain the evolution equations of the multiple-time correlation functions. We also study the evolution of MTCF within the weak coupling limit and it is shown that the multiple-time correlation function of some observables satisfy the Quantum Regression Theorem (QRT), whereas other correlations do not. We set the conditions under which the correlations satisfy the QRT. We illustrate the theory in two different cases; first, solving an exact model for which the MTCF are explicitly given, and second, presenting the results of a numerical integration for a system coupled with a dissipative environment through a non-diagonal interaction.Comment: Submitted (04 Jul 04

Detection statistics in the micromaser

We present a general method for the derivation of various statistical quantities describing the detection of a beam of atoms emerging from a micromaser. The user of non-normalized conditioned density operators and a linear master equation for the dynamics between detection events is discussed as are the counting statistics, sequence statistics, and waiting time statistics. In particular, we derive expressions for the mean number of successive detections of atoms in one of any two orthogonal states of the two-level atom. We also derive expressions for the mean waiting times between detections. We show that the mean waiting times between de- tections of atoms in like states are equivalent to the mean waiting times calculated from the uncorrelated steady state detection rates, though like atoms are indeed correlated. The mean waiting times between detections of atoms in unlike states exhibit correlations. We evaluate the expressions for various detector efficiencies using numerical integration, reporting re- sults for the standard micromaser arrangement in which the cavity is pumped by excited atoms and the excitation levels of the emerging atoms are measured. In addition, the atomic inversion and the Fano-Mandel function for the detection of de-excited atoms is calculated for compari- son to the recent experimental results of Weidinger et al. [1], which reports the first observation of trapping states.Comment: 26 pages, 11 figure