The observed output of an interferometer is the result of interference among the parts of the input light beam traveling along each possible optical path. In complex systems, writing down all these possible optical paths and computing their cumulative effect can become a difficult task. We present an intuitive graph-based method for solving this problem and calculating electric fields within an interferometric setup, classical and quantum. We show how to associate a weighted directed graph with an interferometer and define rules to simplify these associated graphs. Successive application of the rules results in a final graph containing information on the desired field amplitudes. The method is applied to a number of examples in cavity optomechanics and cavity-enhanced interferometers.
The observed output of an interferometer is the result of interference among the parts of the input light beam traveling along each possible optical path. In complex systems, writing down all these possible optical paths and computing their cumulative effect can become a difficult task. We present an intuitive graph-based method for solving this problem and calculating electric fields within an interferometric setup, classical and quantum. We show how to associate a weighted directed graph with an interferometer and define rules to simplify these associated graphs. Successive application of the rules results in a final graph containing information on the desired field amplitudes. The method is applied to a number of examples in cavity optomechanics and cavity-enhanced interferometers.
Richard Feynman’s method of path integrals is based on the fundamental assumption that a system starting at a point 𝐴𝐴 and arriving at a point 𝐵𝐵 takes all possible paths from 𝐴𝐴 to 𝐵𝐵, with each path contributing its own (complex) probability amplitude. The sum of the amplitudes over all these paths then yields the overall probability amplitude that the system starting at 𝐴𝐴 would end up at 𝐵𝐵. We apply Feynman’s method to several optical systems of practical interest and discuss the nuances of the method as well as instances where the predicted outcomes agree or disagree with those of classical optical theory. Examples include the properties of beam-splitters, passage of single photons through Mach-Zehnder and Sagnac interferometers, electric and magnetic dipole scattering, reciprocity, time-reversal symmetry, the optical theorem, the Ewald-Oseen extinction theorem, far field diffraction, and the two-photon interference phenomenon known as the Hong-Ou-Mandel effect.
The photon in the interferometer This chapter introduces many of the ideas of quantum theory by exploring three specific “case studies” of quantum systems. Each is an example of a qubit , a generic name for the simplest type of quantum system. The concepts we develop will be incorporated into a rigorous mathematical framework in the next chapter. Our business here is to provide some intuition about why that mathematical framework is reasonable and appropriate for dealing with the quantum facts of life. Interferometers In Section 1.2 we discussed the two-slit interference experiment with a single photon. In that experiment, the partial waves of probability amplitude were spread throughout the entire region of space beyond the two slits. It is much easier to analyze the situation in an interferometer , an optical apparatus in which the light is restricted to a finite number of discrete beams . The beams may be guided from one point to another, split apart or recombined as needed, and when two beams are recombined into one, the result may show interference effects. At the end of the interferometer, one or more sensors can measure the intensity of various beams. (A beam is just a possible path for the light, so there is nothing paradoxical in talking about a beam of zero intensity.) Figure 2.1 shows the layout of a Mach–Zehnder interferometer , which is an example of this kind of apparatus.
Interference effects are usually observed by intensity measurement. Path indistinguishability by the quantum complementarity principle requires projection of the interfering fields into a common indistinguishable mode before detection. On the other hand, the essence of wave interference is the addition of amplitudes of the interfering fields. Therefore, if amplitudes can be directly measured and added, interference can occur even though the interfering fields are in well-distinguishable modes. Here, we make a comprehensive study in both theory and experiment of a technique by homodyne measurement of field amplitudes to reveal interference. This works for both classical and quantum fields even though there exists distinguishability in the interfering paths of light. This directly challenges the complementarity principle. We present a resolution of this issue from the viewpoint of measurement that emphasizes either particle or wave. This technique is particularly useful for recovering interference in unbalanced interferometers with path imbalance beyond a coherence length of the input field and can be applied to remote sensing to extend the applicable range. Since the amplitude-based interference phenomena studied here are fundamentally different from the traditional intensity-based interference phenomena, our approach leads to a new paradigm to study coherence between optical fields.
The field of Quantum Information Theory provides the theoretical foundation for the pursuit of quantum computers. The ongoing questions of how quantum computers will be realized and what they will achieve, are both very uncertain. However, worldwide efforts are beginning to converge on some answers, and the future of quantum computers is looking brighter than ever. In contribution to the grand goal that is quantum computing, this thesis serves as a demonstration to the usefulness of quantum over classical computing. The central theme of my work, and my collaborators, is the exploration of using quantum systems as a tool for path finding and search algorithms. First, we explore a specific problem of sequential measurements in Quantum Information Theory, namely quantum retrodiction. In experiments involving quantum systems with numerous intermediate interactions, often times the only thing that matters to the experimenters is the final measurement. However, quantum retrodiction is the problem of extrapolating information about the past of a quantum system, by studying the present state. The case studied in this dissertation is the retrodiction of a series of two-outcome measurements, the double and triple qubit-interferometers. In both cases, we are presented with the problem of inferring the results of the two-outcome measurements, using only the final state of the system. Specifically, all of the possible outcomes from the measurements result in a different final state, and we are tasked with distinguishing which final state we have. One version of this problem can be seen as light traversing a series of half-silver mirrors, and we would like to know the path the light took. To distinguish which final state we are given, we employ the use of POVMs (Positive-Operator Valued Measurements) that reflect the symmetry of the system. The effectiveness of these POVMs in solving the retrodiction problem are then presented, along with a few comments as to other types of information that can be extracted. Second, we examine a different type of path finding via a quantum system. For a given graph G that can be expressed by connections and nodes, the same graph can be mapped onto a quantum system with a Hilbert space H. All of the connections in the graph can be represented as states in H, and the nodes act as sites for local unitary operations. The sum of all these local operations give rise to U, which is the unitary operator that acts on H, and describes the behavior of the system. Using this formalism, we explore a particular type of U called a scattering quantum random walk. The quantum random walk is the quantum mechanical analogy to the classical random walk, and serves as a powerful tool for searching on a graph. In this dissertation, we present the case of searching for a marked final node, on a graph that consists of a series of stars. The initial state of the system has an equal probability distribution among all states, reflecting no a priori knowledge about the location of the final node. But by applying the quantum random walk, the final state of the system reaches a superposition where nearly all of the probability is concentrated along the path leading to the marked final node. Lastly, we extend this idea of using a quantum random walk to reveal a path even further. We examine a specific type of graph G, called an “N-tree” graph, and apply our quantum random walk scheme. An N-tree graph can be described as a maze where the solver is presented with a junction consisting of N choices. Beyond each choice, is an identical junction of N choices, and so on. The depth of the maze is categorized by a parameter M, thus, to correctly solve the maze one must make M correct choices in succession. Labeling the marked final nodes as F, we present a quantum algorithm that uses a quantum random walk to locate F faster than any classical algorithm. Specifically, the quantum random walk results in a final state of the system with a high probability concentration on all the states leading to F. Two methods for searching for F are then presented. The first approach mimics the ideology of a Grover Search, relying on repetitive searches until F is found. The second approach presents a novel use of past measurements to dictates how the algorithm proceeds next. Specifically, we take advantage of the fact that states along the path leading to F hold most of the probability in the system. In the end, the second quantum algorithm approach provides an speedup over the classical algorithm of the order O().
In this article we present an experimental proposal for quantum enhanced estimation of optomechanical parameters. The precision of the estimation is improved by using the technique of weak value amplification, which can enlarge the radiation pressure effect of a single-photon on a nano/micro mechanical oscillator. This task is accomplished by using two interferometric setups. Single-photon pulses are sent through one interferometer, producing a maximally path entangled state which drives the cavity optomechanical system. The photons are then postselected in one of the detectors in the output. A second interferometer, whose operation is triggered by every successful postselection, performs an optical measurement of the phase shift generated by the optomechanical system on a classical beam, which encodes the information of the optomechanical parameters. In the presence of fully time-correlated noise, we show that the Fisher information is improved as compared to a standard measurement that employs no postselection.
We present an approach to describe state-of-the-art photonic quantum experiments using graph theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that introducing complex weights in graphs naturally leads to quantum interference. This viewpoint immediately leads to many interesting results, some of which we present here. First, we identify an experimental unexplored multiphoton interference phenomenon. Second, we find that computing the results of such experiments is #P-hard, which means it is a classically intractable problem dealing with the computation of a matrix function Permanent and its generalization Hafnian. Third, we explain how a recent no-go result applies generally to linear optical quantum experiments, thus revealing important insights into quantum state generation with current photonic technology. Fourth, we show how to describe quantum protocols such as entanglement swapping in a graphical way. The uncovered bridge between quantum experiments and graph theory offers another perspective on a widely used technology and immediately raises many follow-up questions.