We show that a quantum observable can be measured by coupling it to a meter which in turn interacts with a reservoir. The complete Hamiltonian is chosen so as to allow for an explicit exact solution of the quantum dynamics. In the continuum limit for the bath the solution displays irreversible behavior, two varieties of which, overdamping and amplification, turn out to be of special relevance. We establish the limit in which a suitable pointer variable (i) behaves effectively classically and (ii) acquires, through the measurement, a probability density of its eigenvalues with well-defined peaks each of which corresponds to one discrete eigenvalue of the measured observable. Under a slightly more restrictive condition the reduced density matrix of the object diagonalizes, during the measurement, in the eigenbasis of the measured observable.
We show that a quantum observable can be measured by coupling it to a meter which in turn interacts with a reservoir. The complete Hamiltonian is chosen so as to allow for an explicit exact solution of the quantum dynamics. In the continuum limit for the bath the solution displays irreversible behavior, two varieties of which, overdamping and amplification, turn out to be of special relevance. We establish the limit in which a suitable pointer variable (i) behaves effectively classically and (ii) acquires, through the measurement, a probability density of its eigenvalues with well-defined peaks each of which corresponds to one discrete eigenvalue of the measured observable. Under a slightly more restrictive condition the reduced density matrix of the object diagonalizes, during the measurement, in the eigenbasis of the measured observable.
We describe the measurement process for a model system whose Schr\"odinger equation yields to explicit solution. Any device for an effectively irreversible measurement must not just be a ``meter'' with a single degree of freedom; it must also contain a ``heat bath'' with many degrees of freedom. Such a bath may cause an effectively instantaneous decay of quantum coherences between macroscopically distinct meter readings, i.e., impart practically classical behavior to the meter. In turn, such a meter can be read without thereby suffering back reactions. We establish the limits in which a meter reading is uniquely related, with reasonable certainty, to a definite eigenvalue of the measured observable.
We present an exact derivation of a process in which a microscopic measured system interacts with the heat-bath and pointer modes of a measuring device, via a coupling involving a general Hermitian operator Λ of the system. In the limit of strong interaction with these modes, over a small time interval, we derive the exact effective many-body density matrix of the measured system plus pointer. We then discuss the interpretation of the dynamics considered as the first stage in the process of quantum measurement, eventually involving the wave-function collapse due to interactions with "the rest of the universe". We establish that the effective density matrix represents the required framework for the measured system and the pointer part of the measuring device to evolve into a statistical mixture described by direct-product states such that the system is in each eigenstate of Λ with the correct quantum-mechanical probability, whereas the expectation values of the pointer-space operators retain amplified information of the system's eigenstate.
By considering the measurement of a quantum spin by two different models of the measuring apparatus, we try to throw some light on the nature of the apparatus that can perform quantum measurement. Our scheme is to consider a quantum spin in interaction with another quantum system playing the role of an apparatus, which in turn is dissipatively coupled to an environment. The coupling with the environment drives the density matrix of the apparatus to a diagonal form. However, measurement is performed only when the reduced density matrix of the apparatus contains correlations with the states of the spin. The two examples considered here show that this happens when the quantum apparatus has a well-defined classical limit. If the apparatus does not have a well-defined classical limit, the reduced density matrix of the apparatus is not correlated with the states of the measured system.
A specific measurement model based on quantum nondemolition monitoring of oscillator quanta is used to illustrate the essential elements of the quantum theory of measurement, viz., state reduction and existence of a ``pointer basis,'' recently discussed by Zurek. The form of the interaction between the meter and its environment determines the system observable recorded by the meter. In our model this quantity is the square of the oscillator's quanta. The corresponding pointer basis states are the meter coherent states. The more accurate the measurement the more excited (and classical) these states become. We show explicitly that the state reduction of the system during measurement is due to nonunitary evolution resulting from the meter-environment coupling. A particular realization of the environment as a photoelectron counter is discussed.
The consensus regarding quantum measurements rests on two statements: (i) von Neumann’s standard quantum measurement theory leaves undetermined the basis in which observables are measured, and (ii) the environmental decoherence of the measuring device (the “meter”) unambiguously determines the measuring (“pointer”) basis. The latter statement means that the environment monitors (measures) selected observables of the meter and (indirectly) of the system. Equivalently, a measured quantum state must end up in one of the “pointer states” that persist in the presence of the environment. We find that, unless we restrict ourselves to projective measurements, decoherence does not necessarily determine the pointer basis of the meter. Namely, generalized measurements commonly allow the observer to choose from a multitude of alternative pointer bases that provide the same information on the observables, regardless of decoherence. By contrast, the measured observable does not depend on the pointer basis, whether in the presence or in the absence of decoherence. These results grant further support to our notion of Quantum Lamarckism, whereby the observer’s choices play an indispensable role in quantum mechanics.
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.