Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as, by an ideal photodetector. Here we derive ͑non-Markovian͒ quantum trajectories for realistic photodetection, including the effects of efficiency, dead time, bandwidth, electronic noise, and dark counts. We apply our theory to a realistic cavity QED scenario and investigate the impact of such detector imperfections on the conditional evolution of the system state. A practical theory of quantum trajectories with realistic detection will be essential for experimental and technological applications of quantum feedback in many areas. The limited utility of quantum-measurement theory as axiomatized by von Neumann ͓1͔ for describing practical laboratory measurements has necessitated the development of more general measurement theories ͓2,3͔. In the past decade the application of such theories has become widespread in quantum optics, in particular for describing continuous monitoring of the photoemission from radiatively damped open systems. They describe the evolution of the conditioned system state in terms of quantum jumps ͓4 -6͔ for direct detection and quantum diffusion ͓6,7͔ for dyne detection. The stochastic evolution equation, termed a quantum trajectory, has also been applied in mesoscopic electronics ͓8͔.Thus far, the main practical utility of quantum-trajectory theory has been in improving the computational efficiency of simulations used to compare models with experimental data. But it is now gaining increasing importance as the quantum generalization of Kalman filtering, which provides essential signal-processing methods in classical estimation, communication, and control engineering. Quantum trajectory theory should, in principle, play the same pivotal role for emerging quantum analogs of these technologies ͓9-11͔. Before this can happen it is essential that the theory be extended to account for the imperfections of realistic measurement devices, as nonideal detector dynamics can dramatically affect the proper inference from measured signals to the conditional quantum state of an observed system.In this paper we present the theory of quantum trajectories for realistic photodetection. We model both photon counters and photoreceivers ͑for homodyne detection͒ and include the effects of efficiency, dead time, bandwidth, electronic noise, and dark counts. The proper treatment of bandwidth limitations and electronic noise are of particular significance as these imperfections are inevitable and predominant concerns in any practical context. They are of central importance in the current generation of experiments on quantum-limited measurement in atomic ͓12͔ and condensed matter ͓13͔ systems.Our theory works by embedding the system within a supersystem that obeys a Markovian equation. If the set of ͑classical͒ detector states is S, then the supersystem is described by the set ͕ s :s S͖. Here Tr͓ s ͔ is the probability that the apparatus is in state s, and s /Tr͓ s ͔ is the sys...

Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. In practice an experimenter has access to an output filtered through various electronic devices, rather than the microscopic states of the detector. This introduces several imperfections into the measurement process, of which only inefficiency has previously been incorporated into quantum trajectory theory. However, all electronic devices have finite bandwidths, and the consequent delay in conveying the output signal to the observer implies that the evolution of the conditional state of the quantum system must be non-Markovian. We present a general method of describing this evolution and apply it to avalanche photodiodes (APDs) and to photoreceivers. We include the effects of efficiency, dead time, bandwidth, electronic noise, and dark counts. The essential idea is to treat the quantum system and classical detector jointly, and to average over the latter to obtain the conditional quantum state. The significance of our theory is that quantum trajectories for realistic detection are necessary for sophisticated approaches to quantum feedback, and our approach could be applied in many areas of physics.

We present a new model for the continuous measurement of a coupled quantum dot charge qubit. We model the effects of a realistic measurement, namely adding noise to, and filtering, the current through the detector. This is achieved by embedding the detector in an equivalent circuit for measurement. Our aim is to describe the evolution of the qubit state conditioned on the macroscopic output of the external circuit. We achieve this by generalizing a recently developed quantum trajectory theory for realistic photodetectors [P. Warszawski, H. M. Wiseman and H. Mabuchi, Phys. Rev. A 65 023802 (2002)] to treat solid-state detectors. This yields stochastic equations whose (numerical) solutions are the "realistic quantum trajectories" of the conditioned qubit state. We derive our general theory in the context of a low transparency quantum point contact. Areas of application for our theory and its relation to previous work are discussed.

In the preceding paper [Warszawski and Wiseman] we presented a general formalism for determining the state of a quantum system conditional on the output of a realistic detector, including effects such as a finite bandwidth and electronic noise. We applied this theory to two sorts of photodetectors: avalanche photodiodes and photoreceivers. In this paper we present simulations of these realistic quantum trajectories for a cavity QED scenario in order to ascertain how the conditioned state varies from that obtained with perfect detection. Large differences are found, and this is manifest in the average of the conditional purity. Simulation also allows us to comprehensively investigate how the quality of the the photoreceiver depends upon its physical parameters. In particular, we present evidence that in the limit of small electronic noise, the photoreceiver quality can be characterized by an effective bandwidth, which depends upon the level of electronic noise and the filter bandwidth. We establish this result as an appropriate limit for a simpler, analytically solvable, system. We expect this to be a general result in other applications of our theory.

Feedback in compound quantum systems is effected by using the output from one subsystem ͑''the system''͒ to control the evolution of a second subsystem ͑''the ancilla''͒ that is reversibly coupled to the system. In the limit where the ancilla responds to fluctuations on a much shorter time scale than does the system, we show that it can be adiabatically eliminated, yielding a master equation for the system alone. This is very significant as it decreases the necessary basis size for numerical simulation and allows the effect of the ancilla to be understood more easily. We consider two types of ancilla: a two-level ancilla ͑e.g., a two-level atom͒ and an infinite-level ancilla ͑e.g., an optical mode͒. For each, we consider two forms of feedback: coherent ͑for which a quantum-mechanical description of the feedback loop is required͒ and incoherent ͑for which a classical description is sufficient͒. We test the master equations we obtain using numerical simulation of the full dynamics of the compound system. For the system ͑a parametric oscillator͒ and feedback ͑intensity-dependent detuning͒ we choose, good agreement is found in the limit of heavy damping of the ancilla. We discuss the relation of our work to previous work on feedback in compound quantum systems, and also to previous work on adiabatic elimination in general.

In the field of atom optics, the basis of many experiments is a two-level atom coupled to a light field. The evolution of this system is governed by a master equation. The irreversible components of this master equation describe the spontaneous emission of photons from the atom. For many applications, it is necessary to minimize the effect of this irreversible evolution. This can be achieved by having a far detuned light field. The drawback of this regime is that making the detuning very large makes the time step required to solve the master equation very small, much smaller than the time scale of any significant evolution. This makes the problem very numerically intensive. For this reason, approximations are used to simulate the master equation, which are more numerically tractable to solve. This paper analyzes four approximations: The standard adiabatic approximation, a more sophisticated adiabatic approximation ͑not used before͒, a secular approximation, and a fully quantum dressed-state approximation. The advantages and disadvantages of each are investigated with respect to accuracy, complexity, and the resources required to simulate. In a parameter regime of particular experimental interest, only the sophisticated adiabatic and dressed-state approximations agree well with the exact evolution.

A D-dimensional Markovian open quantum system will undergo stochastic evolution which preserves pure states, if one monitors without loss of information the bath to which it is coupled. If a finite ensemble of pure states satisfies a particular set of constraint equations then it is possible to perform the monitoring in such a way that the (discontinuous) trajectory of the conditioned system state is, at all long times, restricted to those pure states. Finding these physically realisable ensembles (PREs) is typically very difficult, even numerically, when the system dimension is larger than 2. In this paper, we develop symmetry-based techniques that potentially greatly reduce the difficulty of finding a subset of all possible PREs. The two dynamical symmetries considered are an invariant subspace and a Wigner symmetry. An analysis of previously known PREs using the developed techniques provides us with new insights and lays the foundation for future studies of higher dimensional systems.

We propose a protocol for quantum state tomography of nonclassical states in optomechanical systems. Using a parametric drive, the procedure overcomes the challenges of weak optomechanical coupling, poor detection efficiency, and thermal noise to enable high efficiency homodyne measurement. Our analysis is based on the analytic description of the generalized measurement that is performed when optomechanical position measurement competes with thermal noise and a parametric drive. The proposed experimental procedure is numerically simulated in realistic parameter regimes, which allows us to show that tomographic reconstruction of otherwise unverifiable nonclassical states is made possible.