All-optical versus electro-optical quantum-limited feedback

All-optical feedback can be effected by putting the output of a source cavity through a Faraday isolator and into a second cavity which is coupled to the source cavity by a nonlinear crystal. If the driven cavity is heavily damped, then it can be adiabatically eliminated and a master equation or quantum Langevin equation derived for the first cavity alone. This is done for an input bath in an arbitrary state, and for an arbitrary nonlinear coupling. If the intercavity coupling involves only the intensity (or one quadrature) of the driven cavity, then the effect on the source cavity is identical to that which can be obtained from electro-optical feedback using direct (or homodyne) detection. If the coupling involves both quadratures, this equivalence no longer holds, and a coupling linear in the source amplitude can produce a nonclassical state in the source cavity. The analogous electro-optic scheme using heterodyne detection introduces extra noise which prevents the production of nonclassical light. Unlike the electro-optic case, the all-optical feedback loop has an output beam (reflected from the second cavity). We show that this may be squeezed, even if the source cavity remains in a classical state.Comment: 21 pages. This is an old (1994) paper, but one which I thought was worth posting because in addition to what is described in abstract it has: (1) the first formulation (to my knowledge) of quantum trajectories for an arbitrary (i.e. squeezed, thermal etc.) broadband bath; (2) the prediction of a periodic modification to the detuning and damping of an oscillator for the simplest sort of all-optical feedback (i.e. a mirror) as seen in the recent experiment "Forces between a Single Atom and Its Distant Mirror Image", P. Bushev et al, Phys. Rev. Lett. 92, 223602 (2004

Adiabatic Elimination in Compound Quantum Systems with Feedback

Feedback in compound quantum systems is effected by using the output from one sub-system (``the system'') to control the evolution of a second sub-system (``the ancilla'') which 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 atom) and an infinite-level ancilla (e.g. an optical mode). For each, we consider two forms of feedback: coherent (for which a quantum mechanical description of the feedback loop is required) and incoherent (for which a classical description is sufficient). We test the master equations we obtain using numerical simulation of the full dynamics of the compound system. For the system (a parametric oscillator) and feedback (intensity-dependent detuning) 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.Comment: 18 pages, 12 figures including two subplots as jpeg attachment

Student transitions to blended learning: an institutional case study

This paper examines the experiences of students transitioning to blended learning in the University of Glasgow as part of the QAA Enhancement Themes work on Student Transitions. We draw here on exploratory, qualitative research to examine the benefits, challenges and skills developed by students during transitions to blended learning as a means of advancing understanding, and informing future curriculum design. Data from home undergraduate and international postgraduate students were collected over two years through focus groups, individual interviews and end-of-course quality assurance surveys. We found that while home/undergraduate and international/postgraduate students have similar transition experiences, international taught postgraduates encounter additional challenges in terms of acclimatising to UK higher education (HE), especially within shorter programmes of study and where pedagogical and language differences exist. The findings are integrated in a conceptual framework highlighting the importance of access, acculturation (attitudes) and attributes (skills) to enable learner autonomy to engage effectively in blended learning. The findings have implications for institutional infrastructure, curriculum design and learner development. Further research is required to collect a larger data set as a means of developing the study’s conceptual framework, in order to better understand and support diverse student transitions to blended learning

Phase measurements at the theoretical limit

It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R.B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as n^{-1.5}, where n is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as n^{-2}. It has been shown that a lower limit to the phase variance that is introduced scales as ln(n)/n^2. Here we introduce an adaptive technique that attains this theoretical lower limit.Comment: 9 pages, 5 figures, updated with better feedback schem

Robust unravelings for resonance fluorescence

Monitoring the fluorescent radiation of an atom unravels the master equation evolution by collapsing the atomic state into a pure state which evolves stochastically. A robust unraveling is one that gives pure states that, on average, are relatively unaffected by the master equation evolution (which applies once the monitoring ceases). The ensemble of pure states arising from the maximally robust unraveling has been suggested to be the most natural way of representing the system [H.M. Wiseman and J.A. Vaccaro, Phys. Lett. A {\bf 250}, 241 (1998)]. We find that the maximally robust unraveling of a resonantly driven atom requires an adaptive interferometric measurement proposed by Wiseman and Toombes [Phys. Rev. A {\bf 60}, 2474 (1999)]. The resultant ensemble consists of just two pure states which, in the high driving limit, are close to the eigenstates of the driving Hamiltonian $\Omega\sigma_{x}/2$. This ensemble is the closest thing to a classical limit for a strongly driven atom. We also find that it is possible to reasonably approximate this ensemble using just homodyne detection, an example of a continuous Markovian unraveling. This has implications for other systems, for which it may be necessary in practice to consider only continuous Markovian unravelings.Comment: 12 pages including 5 .eps figures, plus one .jpg figur

Complete parameterization, and invariance, of diffusive quantum trajectories for Markovian open systems

The state matrix $\rho$ for an open quantum system with Markovian evolution obeys a master equation. The master equation evolution can be unraveled into stochastic nonlinear trajectories for a pure state $P$, such that on average $P$ reproduces $\rho$. Here we give for the first time a complete parameterization of all diffusive unravelings (in which $P$ evolves continuously but non-differentiably in time). We give an explicit measurement theory interpretation for these quantum trajectories, in terms of monitoring the system's environment. We also introduce new classes of diffusive unravelings that are invariant under the linear operator transformations under which the master equation is invariant. We illustrate these invariant unravelings by numerical simulations. Finally, we discuss generalized gauge transformations as a method of connecting apparently disparate descriptions of the same trajectories by stochastic Schr\"odinger equations, and their invariance properties.Comment: 10 pages, including 5 figures, submitted to J. Chem Phys special issue on open quantum system

Squeezing via feedback

We present the quantum theory of optical cavity feedback mediated by homodyne detection, with an arbitrary time delay. We apply this theory to a system with nonclassical dynamics, a sub-Poissonian pumped laser. By using the feedback to phase lock the laser it is possible to produce output light which exhibits perfect quadrature squeezing on resonance, rather than just sub-Poissonian intensity statistics. However, we also show that feedback mediated by homodyne detection (or any other extracavity measurement) cannot produce nonclassical light unless the cavity dynamics can do so without feedback. Furthermore, in systems which already exhibit squeezing, such feedback can only degrade the squeezing in the output. With feedback mediated by an intracavity measurement, these theorems do not apply. We show that an (admittedly unrealistic) intracavity quantum nondemolition quadrature measurement allows arbitrary squeezing to be produced by controlling the amplitude of a coherent driving field

Feedback-control of quantum systems using continuous state-estimation

We present a formulation of feedback in quantum systems in which the best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system. We apply this method to the problem of cooling and confining a single quantum degree of freedom, and compare it to current schemes in which the measurement signal is fed back directly in the manner usually considered in existing treatments of quantum feedback. Direct feedback may be combined with feedback by estimation, and the resulting combination, performed on a linear system, is closely analogous to classical LQG control theory with residual feedback.Comment: 12 pages, multicol revtex, revised and extende

Quantum Kinetic Theory III: Quantum kinetic master equation for strongly condensed trapped systems

We extend quantum kinetic theory to deal with a strongly Bose-condensed atomic vapor in a trap. The method assumes that the majority of the vapor is not condensed, and acts as a bath of heat and atoms for the condensate. The condensate is described by the particle number conserving Bogoliubov method developed by one of the authors. We derive equations which describe the fluctuations of particle number and phase, and the growth of the Bose-Einstein condensate. The equilibrium state of the condensate is a mixture of states with different numbers of particles and quasiparticles. It is not a quantum superposition of states with different numbers of particles---nevertheless, the stationary state exhibits the property of off-diagonal long range order, to the extent that this concept makes sense in a tightly trapped condensate.Comment: 3 figures submitted to Physical Review

Optomechanical scheme for the detection of weak impulsive forces

We show that a cooling scheme and an appropriate quantum nonstationary strategy can be used to improve the signal to noise ratio for the optomechanical detection of weak impulsive forces.Comment: 4 pages, Revtex, 1 figur