We consider a quantum particle on a one dimensional lattice subject to weak local measurements and study its stochastic dynamics conditioned on the measurement outcomes. Depending on the measurement strength our analysis of the quantum trajectories reveals dynamical regimes reach- ing from quasi-coherent wave packet oscillations to a Zeno-type dynamics. We analyse how these dynamical regimes are directly reflected in the spectral properties of the noisy measurement records.
We consider a quantum particle on a one dimensional lattice subject to weak local measurements and study its stochastic dynamics conditioned on the measurement outcomes. Depending on the measurement strength our analysis of the quantum trajectories reveals dynamical regimes reach- ing from quasi-coherent wave packet oscillations to a Zeno-type dynamics. We analyse how these dynamical regimes are directly reflected in the spectral properties of the noisy measurement records.
We consider a qubit governed by a sequence of weak measurements, with the measurement strength modified in a time- and state-dependent manner. Here we show that the resulting trajectory of the qubit in the phase space can be weakly controlled without any direct action on the qubit (control-free control), even if only one fixed observable is measured. In particular, we demonstrate a possibility of a weak form of a stochastic ratchet, allowing us to weakly push the qubit's state without any net force in the effective potential. Furthermore, if the weak measurement strength is significantly reduced in a way conditioned to some particular state, a dynamical localization near this state takes place. If the measurement strength is reduced to zero, a singularity appears, which behaves like an artificial basis state.
Abstract We investigate the quantum Zeno effect (QZE) in spin 1/2, spin 1 and spin 3/2 open quantum systems undergoing Rabi oscillations, revealing unexplored features for the spin 1 and spin 3/2 systems. The systems interact with an environment designed to perform continuous measurements of an observable, driving the systems stochastically towards one of the eigenstates of the corresponding operator. The system-environment coupling constant represents the strength of the measurement. Stochastic quantum trajectories are generated by unravelling a Markovian Lindblad master equation using the quantum state diffusion formalism. These are regarded as a more appropriate representation of system behaviour than consideration of the averaged evolution since the latter can mask the effect of measurement. Complete positivity is maintained and thus the trajectories can be considered as physically meaningful. The QZE is investigated over a range of measurement strengths. Increasing the strength leads to greater system dwell in the vicinity of the eigenstates of the measured observable and lengthens the time taken by the system to return to that eigenstate, thus the QZE emerges. For very strong measurement, the Rabi oscillations resemble randomly occurring near-instantaneous jumps between eigenstates. The trajectories followed by the quantum system are heavily dependent on the measurement strength which other than slowing down and adding noise to the Rabi oscillations, changes the paths taken in spin phase space from a circular precession into elaborate figures-of-eight. For spin 1 and spin 3/2 systems, the measurement strength determines which eigenstates are explored and the QZE is stronger when the system dwells in the vicinity of certain eigenstates compared to others.
This thesis considers quantum systems which interact with their surrounding environment. Specifically we consider the underlying stochastic dynamics which arise from the back-action of measurements performed in the environment. These quantum trajectories can be tracked by continuously monitoring the environment, as is now possible in modern experiments. We consider phenomena understood at the average level of the system state density matrix, and explore these at this more information rich level. This is especially relevant for experimental setups and implementation of quantum technologies. Firstly, we consider how systems exhibiting metastability behave at the level of stochastic quantum trajectories. In doing so we make comparisons to classical theories of metastability, where trajectories exhibit fast relaxation into distinct phases and slow transitions between them. We also illustrate when this phenomenology does not hold, with instead the emergence of a slow manifold in the form of a decoherence free subspace. We also highlight the useful role that quantum reset processes can play in the analysis of stochastic quantum trajectories. The dynamics of the stochastic quantum trajectories depends on the choice of environmental measurements. There are many choices which leave the average dynamics unchanged, but give different statistical ensembles of quantum trajectories. This gauge freedom of the average dynamics is well known. We derive the analogous result at the level of quantum trajectory ensembles; giving the measurement protocols which leave the generator of the stochastic trajectory dynamics unchanged. We also extend this to include joint trajectories of conditional system states and their associated measurement records, either full records or coarse-grained. Using these results on gauge freedoms of quantum trajectories, we consider the conditions under which the trajectory dynamics exhibit a weak unitary symmetry. We then consider under what conditions the joint state of the system and environment exhibit symmetric dynamics, including under various levels of dephasing. We find that the joint dynamics always features a separable symmetry directly related to that of the quantum master equation, but for quantum trajectories the corresponding symmetry is present only if the counting measurement satisfies certain conditions.
This is a theoretical thesis in the area of quantum measurement theory. Due to the extensive breadth of this field we choose to narrow our focus to examine a particular problem - the quantum Zeno effect (defined below). Quantum measurement theory is introduced in Chap. 1, using the terminology of effects and operations. This approach allows an operational definition of such terms as a state vector, an ensemble, and a measurement device (for instance), and a consideration of interactions between quantum systems and inaccurate measurement devices. We further introduce the quantum trajectories approach to consider the evolution of an individual quantum system subject to measurement. The quantum Zeno effect is introduced in Chap. 2. Any quantum treatment of a measurement interaction must consider the measurement backaction onto the measured system and this backaction will disrupt the free evolution of the system. The quantum Zeno effect occurs in the strong measurement limit where the measurement backaction totally freezes the evolution of the system, thus rendering the measurement useless. The effect is introduced via projective measurements of two level systems subject to measurement of level populations. At this stage we are able to discuss the main questions addressed by this thesis, and present its structure in Chap. 2. We then develop a new measurement model for the interaction between a system and a measurement device in Chap. 3. Our motivation in doing this is to better model the usual laboratory meter, and in our approach the meter dynamics are such that it relaxes towards an appropriate readout of the system parameter of interest. The irreducible quantum noise of the meter introduces fluctuations that drive the stochastic dynamical collapse of the system wavefunction. In our model, the measured system dynamics (if treated selectively) are described by a stochastic, nonlinear Schroedinger equation. A double well system subject to position measurement provides a natural first application for this model. This is done in Chap. 4 where we monitor the coherent tunnelling of a particle from one well to the other. The advantage afforded by considering this system is that it displays differing regimes where the measurement observable (position) is approximated as possessing either, respectively, a continuous or a discrete eigenvalue structure. Thus, we use this one model to explore the quantum Zeno effect in both measurement regimes. The above treatment is of a theoretical measurement model. In Chap. 5 we turn to consider a recent experimental test of the quantum Zeno effect which examined the dynamics of a two level atom subject to pulsed measurements of atomic level populations. We treat a slightly modified experiment in a fully continuous measurement regime. By first unravelling the optical Bloch equations, and second, using the quantum trajectories approach we demonstrate the existence of certain measurement regimes where there is a quantum Zeno effect, and other regimes where no measurement of the atomic populations is being effected at all. Through these results we demonstrate the importance of making a full analysis of the system-detector interaction before any conclusions can be made. In the remainder of the thesis we propose further possible tests of the quantum Zeno effect. In Chap. 6 the evolution of a Rydberg atom exchanging one photon with a single cavity mode subject to measurement is examined. The measurement is made by monitoring the photon number occupancy of the cavity mode using a beam of Rydberg atoms configured so as to perform phase sensitive detection. In the limit of frequent monitoring we show that the free oscillation of the atomic inversion is disrupted, and the atom is trapped close to its initial state. This is the quantum Zeno effect. In Chap. 7 we realize the Zeno effect on two possible systems. We consider first, a two level Jaynes-Cumming atom interacting with a cavity mode, and second, two electromagnetic modes configured as a multi-level parametric frequency converter. These systems interact with another cavity mode via a quadratic coupling system based on four wave mixing, and constructed to be a quantum nondemolition measurement of the photon number. This mode is damped to the environment thus effecting a measurement of the system populations. Again we show that this interaction, can manifest the quantum Zeno effect. Our explicit modelling of the system-detector interaction enables us to show how the effect depends on the resolution time of the detector. Finally, we consider a proposed measurement of the square of the quadrature phase of an electromagnetic mode in Chap. 8. Here, a three mode interaction mediated by a second order nonlinear susceptibility is considered. One mode, the pump, is prepared in a feedback generated photon number state to give insight into the role of pump noise. The other two modes are treated as an angular momentum system, and we show that photon counting on the two mode rotation system effects the above mentioned measurement. In addition, this measurement provides a direct measure of the second order squeezing of the signal. With that we finish our investigation of the quantum Zeno effect using the techniques of quantum measurement theory. However, in the epilogue [Chap. 9] we note that no thesis in quantum measurement theory would be complete without some consideration of the ``meaning attributed to the theory. In the epilogue we take a novel historical approach and examine the method by which metaphysical theories are formed to draw conclusions regarding quantum metaphysics.
In the quantum Zeno effect, quantum measurements can block the coherent oscillation of a two level system by freezing its state to one of the measurement eigenstates. The effect is conventionally controlled by the measurement frequency. Here we study the development of the Zeno regime as a function of the measurement strength for a continuous partial measurement. We show that the onset of the Zeno regime is marked by a cascade of transitions in the system dynamics as the measurement strength is increased. Some of these transitions are only apparent in the collective behavior of individual quantum trajectories and are invisible to the average dynamics. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These predicted dynamical features, which can be readily observed in current experiments, show the coexistence of fundamentally unpredictable quantum jumps with those continuously monitored and reverted in recent experiments.