Motivated by decoherence theory, a heuristic formulation of quantum measurement theory is produced in which the macroscopic measuring instrument is explicitly included on the same footing as the quantum system being measured. The theory takes the form of a master equation for the joint density matrix of the measuring instrument and quantum system. The master equation refers to both systems in a symmetric way. Solution of the master equation using quantum trajectory theory furnishes a quasi-Hamiltonian evolution of the measuring process, punctuated by quantum jumps involving both the atomic system and the states of the measuring instrument. The projection postulate of quantum measurement theory emerges as a result from the formalism. A pervasive quantum Zeno effect and delayed detection are two phenomena, usually masked by the standard treatment of quantum measurement theory, which appear in this formalism.
Motivated by decoherence theory, a heuristic formulation of quantum measurement theory is produced in which the macroscopic measuring instrument is explicitly included on the same footing as the quantum system being measured. The theory takes the form of a master equation for the joint density matrix of the measuring instrument and quantum system. The master equation refers to both systems in a symmetric way. Solution of the master equation using quantum trajectory theory furnishes a quasi-Hamiltonian evolution of the measuring process, punctuated by quantum jumps involving both the atomic system and the states of the measuring instrument. The projection postulate of quantum measurement theory emerges as a result from the formalism. A pervasive quantum Zeno effect and delayed detection are two phenomena, usually masked by the standard treatment of quantum measurement theory, which appear in this formalism.
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.
We show that all nonrelativistic quantum processes, whether open or closed, are either unitary or probabilistic unitary, i.e., probabilistic combinations of unitary evolutions. This means that, for open quantum systems, their continuous dynamics can always be described by the Lindblad master equation with all jump operators being unitary. We call this formalism the probabilistic unitary formulation of open quantum system dynamics. This formalism is shown to be exact under all cases, and does not rely on any assumptions other than the continuity and differentiability of the density matrix. Moreover, it requires as few as $d\ensuremath{-}1$ jump operators, instead of ${d}^{2}\ensuremath{-}1$, to describe the open dynamics in the most general case, where d is the dimension of Hilbert space of the system. Importantly, different from the conventional Lindblad master equation, this formalism is state dependent, meaning that the Hamiltonian, jump operators, and rates in general all depend on the current state of the density matrix. Hence one needs to know the explicit expression of the density matrix in order to write down the probabilistic unitary master equation explicitly. Experimentally, the formalism provides a scheme to control a quantum state to evolve along designed nonunitary quantum trajectories, and can be potentially useful in quantum computing and quantum control scenes since only unitary resources are needed for implementation.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr) of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics.
In this paper we derive master equations for two or more systems coupled to each other, perhaps strongly, by using a generalization of the usual projection-operator technique to include time-dependent projection operators. The coupled systems may be either similar or dissimilar and classical or quantum mechanical. Whereas the customary approaches to coupled systems are best able to treat situations in which some of the systems are "baths" with a specified density operator or phase-space probability density, our approach allows us to treat situations where it is necessary or convenient to treat the coupled systems on an equal footing. In our scheme the "relevant" part of the full density operator is considered to be the uncorrelated part of the full density operator and is a symmetric functional of the reduced density operators of each of the coupled subsystems. The "irrelevant" part of the density operator is then the part describing correlations between the coupled systems. Our formalism is particularly useful where systems are coupled to one another predominantly in a self-consistent fashion. First, we develop exact master equations for two coupled systems, taking as our prototype the dynamical problem of quantum optics, where a spatially extended collection of two-level atoms interact with a multimode optical field. We then generalize our results to N coupled systems, taking as our prototype the kinetics of a classical nonideal gas interacting through two-body forces, and derive exact master equations for the system. We then consider as examples several approximate theories resulting from our exact equations. In the case of the imperfect gas we investigate the low-density limit and show how Bogoliubov's form of the Boltzmann equation emerges from our formalism, as well as corrections due to Klimontovich. We consider as special cases of our exact quantum-optical equations the equations in the first Born approximation, with and without memory, and show how several existing quantum-optical master equations are contained in our general results. As a second example in quantum optics, we consider the case where the predominant behavior of the system is described by the self-consistent-field or coupled Bloch and Maxwell equations and derive a first-order perturbation description for deviations from self-consistent-field behavior.
Introduction and overview Much that is studied in quantum measurement theory, and thus in this book, is very different from that studied in its classical counterpart, Bayesian inference. While quantum measurement theory is in one sense a minimal extension of the latter to quantum states, quantum measurements cause dynamical changes that never need appear in a classical theory. This has a multitude of ramifications, many of which you will find in this book. One such consequence is a limit to the information that can be obtained about the state of a quantum system, and this results in relationships between information gained and dynamics induced that do not exist in classical systems. Because of the limits to information extraction, choosing the right kinds of measurements becomes important in a way that it never was for classical systems. Finding the best measurements for a given purpose, and working out how to realize these measurements, is often nontrivial. The relationship between information and disturbance impacts the control of quantum systems using measurements and feedback, and we discuss this application in Chapter 5. The ability to extract information about quantum systems is also important in using quantum systems to make precise measurements of classical quantities such as time, acceleration, and the strengths of electric and magnetic fields. This subject is referred to as quantum metrology, and we discuss it in Chapter 6.