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 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.
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 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.
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-Markovian) 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.
In the preceding paper (Warszawski P and Wiseman H M 2002 J. Opt. B: Quantum Semiclass. Opt. 4 1) 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 photodetector: avalanche photodiodes and photoreceivers. In this paper we present simulations of these realistic quantum trajectories for a cavity quantum electro-dynamics 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 photoreceiver (PR) depends upon its physical parameters. In particular, we present evidence that in the limit of small electronic noise, the PR 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.
To obtain information about a system of interest a measurement has to be made. In experiments that probe the quantum nature of our world, the system itself is, in general, necessarily affected by the act of measurement. If the system is weakly coupled to its bath and the dynamics are such that information concerning the system is spread throughout the many degrees of freedom of the bath, and the bath is being measured then a stochastic master equation for the conditioned state of the system can be found. This is termed a quantum trajectory equation. Realistic detectors are not perfect. Information is lost in the conversion to a signal that the observer can use. This loss may occur in the detector itself, in the circuit containing the detector (described by a response time and electronic noise) or at the circuit output (electronic output noise). In order to obtain a true quantum trajectory for the experiment, the observer must condition the state of the quantum system on results that are available in the laboratory rather than on the microscopic events considered previously in quantum trajectories. A method for treating this was first proposed by Warszawski, Wiseman and Mabuchi [Phys. Rev. A 65, 023802 (2002)], in which the quantum system is embedded within a supersystem that also contains the state of the detector. They applied their theory to photodetectors of various sorts. Warszawski has also done the preliminary work on applying this theory to detecting the state of a pair of quantum dots using a SET (single-electron transistor) [MSc. Thesis, Griffith University (2001)]. The resulting theory is termed 'realistic' quantum trajectory theory. In this thesis, the approach of Warszawski, et al.is applied to various solidstate readout devices. These include the SET, the QPC (quantum point contact), and the RF-QPC (radio-frequency QPC). Numerically obtained realistic quantum trajectories for the QPC agree with heuristic results. In particular, in certain limits, the realistic quantum trajectories can take on the appearance of ideal quantum trajectories. This thesis also resolves a problem in solid-state continuous quantum measurement theory by deriving a quantum trajectory model for a SET-monitored charge qubit, that guarantees physically meaningful qubit states. The particular limit necessary to achieve this is discussed, and the SET measurement quality is analysed using techniques borrowed from quantum optics. Conditions for which the SET can approach operation at the limit allowed by quantum mechanics are given. This is also done for the QPC, for which the results agree with previous work.
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.
Introduction A very general concept of a quantum trajectory would be the path taken by the state of a quantum system over time. This state could be conditioned upon measurement results, as we considered in Chapter 1. This is the sort of quantum trajectory we are most interested in, and it is generally stochastic in nature. In ordinary use, the word trajectory usually implies a path that is also continuous in time. This idea is not always applicable to quantum systems, but we can maintain its essence by defining a quantum trajectory as the path taken by the conditional state of a quantum system for which the unconditioned system state evolves continuously. As explained in Chapter 1, the unconditioned state is that obtained by averaging over the random measurement results which condition the system. With this motivation, we begin in Section 4.2 by deriving the simplest sort of quantum trajectory, which involves jumps (that is, discontinuous conditioned evolution). In the process we will reproduce Lindblad's general form for continuous Markovian quantum evolution as presented in Section 3.6. In Section 4.3 we relate these quantum jumps to photon-counting measurements on the bath for the model introduced in Section 3.11, and also derive correlation functions for these measurement records. In Section 4.4 we consider the addition of a coherent field (the ‘local oscillator’) to the output before detection. In the limit of a strong local oscillator this is called homodyne detection, and is described by a continuous (diffusive) quantum trajectory.
We investigate the effect of imperfections in realistic detectors upon the problem of quantum state and parameter estimation by continuous monitoring of an open quantum system. Specifically, we have reexamined the system of a two-level atom with an unknown Rabi frequency introduced by Gambetta and Wiseman [Phys. Rev. A 64, 042105 (2001)]. We consider only direct photodetection and use the realistic quantum trajectory theory reported by Warszawski, Wiseman, and Mabuchi [Phys. Rev. A 65, 023802 (2002)]. The most significant effect comes from a finite bandwidth, corresponding to an uncertainty in the response time of the photodiode. Unless the bandwidth is significantly greater than the Rabi frequency, the observer's ability to obtain information about the unknown Rabi frequency, and about the state of the atom, is severely compromised. This result has implications for quantum control in the presence of unknown parameters for realistic detectors, and even for ideal detectors, as it implies that most of the information in the measurement record is contained in the precise timing of the detections.