State determination in continuous measurement

There is currently great interest in experiments which obtain useful information about a quantum system or state in single runs of an experiment. Very recently it has been possible to distinguish the quantum mechanical and classical models of the interaction of an atom with a mode of a high finesse optical cavity as the result of continuous monitoring of the output light while a single atom passes through the cavity [1]. As a result of this continuous monitoring the quantum mechanical backaction of the measurement process may be expected to have a significant effect on the evolution of individual runs of the experiment. Moreover, it may be possible in the future to modify the evolution of the system through feedback based on this continuous measurement [2]. Current experimental technology, such as that described in [1], is reaching the point at which determining the state of the system and observing the effects of backaction and feedback in a single run of an experiment is a real possibility. With this situation in mind, we discuss the identification of the state of the system following a period of continuous observation and the extent to which this state can be tracked, taking into account factors such as imperfect initial knowledge of the state and imperfect detection efficiency. We focus on a model of continuous position measurement of a mechanical oscillator which is relevant to the experiment of Mabuchi et al. but which also has relevance to other instances of interferometric position monitoring such as gravitational wave detection. The problem of describing quantum systems undergoing continuous measurement has attracted much theoretical interest in recent years. As discussed by Wiseman [3] these theories admit a variety of interpretations; as tools for efficient stochastic calculation of ensemble averages in lieu of solving master equations [4], as equations describing the evolution of systems conditioned on measurements [5–7] and as a description of the evolution of a system coupled to an environment, in which collapse of the wavefunction is supposed to be associated with the coupling to the environment [8]. Here, we take the second viewpoint namely that the conditioned state represents the observer’s best description of the system state given the results of the continuous measurement process. Adopting the first or third viewpoints one is led to describe the system by a pure state vector throughout the evolution although the reasons for doing so are somewhat different in each case. By contrast, a description of one’s conditioned state of knowledge necessarily requires mixed states in order to account for incomplete knowledge of the system. From this viewpoint the fundamental equation for the conditioned evolution is the stochastic master equation (SME) [9]. This is able to account for the effects of mixed initial states, imperfect detection efficiencies and the existence of unmeasured couplings to the environment. However, to date, relatively little work has attempted to address the evolution of the conditioned state in any of these situations [10]. In this paper we consider a system which is simple enough that almost all the work can be done analytically and which admits a treatment of all of these imperfections. This helps in developing intuition about the role of SME’s and their possible relevance to experiments. A projective measurement has the property that if the result of the measurement is known, the state after the measurement is pure, and depends only on the measurement result. It would be hoped that in a continuous measurement there would be some finite interval of time after which the measurement has effectively given rise to a projection, so that the system is placed in a particular state which depends only on the sequence of measurement results and which can be calculated without knowledge of the initial state. If the resulting state is pure then a stochastic Schrödinger equation (SSE) would be a perfectly adequate tool for describing the subsequent system evolution. In this paper we investigate the conditions which lead to such an effective collapse and over what timescale it takes place. This is made possible by considering density matrices and the SME rather tha wave vectors and the SSE. In a real experiment there will also be uncontrolled, unmeasured couplings of the system to the environment and in this case the effects of the measurement will compete not only with the coherent internal dynamics of the system but also with the