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Novel quantum states and measurements

Samuel L. Braunstein

Published Jan 1, 1988• 1 citation
Publisher

Abstract

i) The mechanics of higher order parametric amplifiers is studied. It is shown that these devices produce meaningful states; numerical calculations are performed that demonstrate the convergence of matrix elements associated with these states. Further, the correspondence between the classical and evolution for these devices is studied and the differences explained by a kind of quantum diffusion. Finally, the possibility of producing ordinary squeezed light with these devices is noted. ii) The generation of squeezed light always involves some scheme that amounts to pumping electromagnetic modes at near twice their natural frequency. When the pump is itself treated mechanically, extra noise is introduced that ultimately limits the amount of squeezing achievable. Detailed calculations are carried out in this regard for the parametric amplifier. It is found that the pump's initial phase noise is responsible for this limit. iii) Quantum-mechanical measurements are usually described by applying the standard rules to a measurement model. They can also be described by a formalism that uses mathematical objects called Effects and Operations. These two descriptions should be equivalent. D'Espagnat has raised a question about the usage of this formalism of Effects and Operations for repeated measurements. This question is cleared up, and the source of the discrepancy is given a simple interpretation. iv) Usually, an inequality that is chained becomes a weaker inequality. Chaining the Bell inequality, however, leads to stronger violations by mechanics. Further, a new kind of Bell inequality, based on the information obtained in a measurement, is derived. This information Bell inequality can be used to formulate tests of local realism in very general circumstances, e.g., higher spin versions of the EPR experiment. These new inequalities yield an interpretation for the size of their violation and lead to the formulation of a hierarchy of Bell inequalities for which two-particle Bell inequalities play a special role.

DOI
10.7907/9k1b-7x39.

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i) The mechanics of higher order parametric amplifiers is studied. It is shown that these devices produce meaningful states; numerical calculations are performed that demonstrate the convergence of matrix elements associated with these states. Further, the correspondence between the classical and evolution for these devices is studied and the differences explained by a kind of quantum diffusion. Finally, the possibility of producing ordinary squeezed light with these devices is noted.
ii) The generation of squeezed light always involves some scheme that amounts to pumping electromagnetic modes at near twice their natural frequency. When the pump is itself treated mechanically, extra noise is introduced that ultimately limits the amount of squeezing achievable. Detailed calculations are carried out in this regard for the parametric amplifier. It is found that the pump's initial phase noise is responsible for this limit.
iii) Quantum-mechanical measurements are usually described by applying the standard rules to a measurement model. They can also be described by a formalism that uses mathematical objects called Effects and Operations. These two descriptions should be equivalent. D'Espagnat has raised a question about the usage of this formalism of Effects and Operations for repeated measurements. This question is cleared up, and the source of the discrepancy is given a simple interpretation.
iv) Usually, an inequality that is chained becomes a weaker inequality. Chaining the Bell inequality, however, leads to stronger violations by mechanics. Further, a new kind of Bell inequality, based on the information obtained in a measurement, is derived. This information Bell inequality can be used to formulate tests of local realism in very general circumstances, e.g., higher spin versions of the EPR experiment. These new inequalities yield an interpretation for the size of their violation and lead to the formulation of a hierarchy of Bell inequalities for which two-particle Bell inequalities play a special role.
In this work we look for novel classes of Bell's inequalities and methods to produce them. We also find their quantum violations including, if possible, the maximum one. The Jordan bases method that we explain in Chapter 2 is about using a pair of certain type of orthonormal bases whose spans are subspaces related to measurement outcomes of incompatible quantities on the same physical system. Jordan vectors are the briefest way of expressing the relative orientation of any two subspaces. This feature helps us to reduce the dimensionality of the parameter space on which we do searches for optimization. The work is published in [24]. In Chapter 3, we attempt to find a connection between group theory and Bell's theorem. We devise a way of generating terms of a Bell's inequality that are related to elements of an algebraic group. The same group generates both the terms of the Bell's inequality and the observables that are used to calculate the quantum value of the Bell expression. Our results are published in [25][26]. In brief, Bell's theorem is the main tool of a research program that was started by Einstein, Podolsky, Rosen [19] and Bohr [8] in the early days of quantum mechanics in their discussions about the core nature of physical systems. These debates were about a novel type of physical states called superposition states, which are introduced by quantum mechanics and manifested in the apparent inevitable randomness in measurement outcomes of identically prepared systems. Bell's huge contribution was to find a means of quantifying the problem and hence of opening the way to experimental verification by rephrasing the questions as limits on certain combinations of correlations between measurement results of spatially separate systems [7]. Thanks to Bell, the fundamental questions related to the nature of quantum mechanical systems became quantifiable [6]. According to Bell's theorem, some correlations between quantum entangled systems that involve incompatible quantities are not allowed by classical mechanics, a feature that is called as "quantum nonlocality". An experimental observation of those correlations, in other words, a violation of the limits imposed by classical physics, implies the correctness of quantum description and invalidates the classical, local realistic models. The first Bell experiments were proposed by Clauser, Horne, Shimony, and Holt, who invented the most famous Bell's inequality [13]. Later, the Aspect experiments were satisfactory enough for the physics community to be conclusive about the validation of quantum mechanics [1][3][4][2]. Ekert's work on applications of quantum nonlocality to communication resulted in the new field of quantum communication and cryptography, and turned the research program into a practical one [20]. Pitowsky showed a method to find all expressions of limitations due to local realism, all Bell's inequalities, for a given physical scenario. He also proved that the problem is, unfortunately, NP-complete and hence as the scenarios get more complex, they also become computationally intractable [33][34]. Therefore, different methods for the solution of special cases of the problem are necessary. Inequalities found for those special cases can be called classes of Bell's inequalities. For example, Werner and Wolf [41] and Collins, Gisin, Linden, Massar, and Popescu [16] found classes that cover a wide range of scenarios. Our work is a similar kind of effort to produce and study new types of Bell's inequalities.
This paper demonstrates the capability to simulate EPR experiments in a realistic way (i.e.without transmission of information between the source and the detector) and confirms even more the Bell's theorem, by demonstrating that effectively non-local variables are required to violate the Bell's inequality.Unlike all EPR experiments, this modeling instead of simply noting the inadequacy of a local modeling proposes a new way of non-local idealization.These simulations work on spreadsheet by idealizing objects with a non-linear notion, the extended object.This allows to set up a new principle of locality (or non-locality) and a new meaning on the loopholes of EPR experiments.It sheds new light on several concepts of Quantum Mechanics (QM): the quantum states (would represent an equivalence class of many different indistinguishable objects seen as equivalent in terms of measurement), the wave function (would represent a probability on this equivalence class seen as a single set despite different probabilities of happening for the individual objects composing this equivalence class), the measurement (in our simulation, the measured object is not modified), the superposition of states (in our simulation, Schrödinger's cat is never both dead and alive).These simulations lead to several kinds of EPR experiments with violations of the Bell's inequalities with values different from the QM, allow to realize entanglements with more than 2 objects (whatever the number of Alice and Bob) or even with more than 2 results (beyond the only "+1" or "-1" outputs).Many questions remain to be explored in the physics domain on quantum phenomena (teleportation, encryption, quantum computing, etc.), but this theoretical approach also reveals the necessity of developing new non-linear tools in mathematics domain and seems to show that QM could finally be a limit case of a more fundamental nonlinear physics theory founded on these How to cite this paper: Le Corre, S.
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
This thesis is primarily a theoretical study of degenerate parametric amplification as a means of generating squeezed-state light. i) A wideband traveling-wave formalism is developed for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations — spatial Langevin equations — that propagate temporal Fourier components of the field through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion, and pump quantum fluctuations. The dominant effects of dispersion and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier are analyzed. ii) The wideband formalism of i) is used to carry out a more detailed analysis of the effects of phase mismatching. With the assumption of a lossless medium and a classical pump, we find that parametric amplification is capable of generating squeezed-state light over a wide band if materials with large χ(2) nonlinearities can be found, and that the squeezing bandwidth can be enhanced by phase mismatching away from degeneracy. iii) We consider again the effect of pump quantum fluctuations on the squeezing produced by parametric amplification. We perform discrete-mode calculations for a parametric amplifier with a quantum pump, and discuss some of the limitations of calculations of this sort in quantum optics. We derive stochastic differential equations (SDEs) for one- and two-mode parametric amplifiers, and from them obtain an iterative solution showing that pump quantum fluctuations impose a limitation on the degree of squeezing obtainable from a parametric amplifier. iv) A possible application of squeezing is considered; in particular, we study the effects of squeezing the intracavity noise in a laser oscillator. We solve the classical noise problem of a realistic laser model by making a bold — and possibly unrealizable — assumption, that the in-phase and quadrature Langevin sources which are responsible for the "noisiness" of the laser can be squeezed. We show that the effect of squeezing the in-phase quadrature is to reduce the phase noise, including the linewidth, of the laser but, due to amplitude-phase coupling, not to eliminate them altogether.
Bell's theorem applies to the normalizable approximations of original Einstein-Podolsky-Rosen (EPR) state. The constructions of the proof require measurements difficult to perform, and dichotomic observables. By noticing the fact that the four mode squeezed vacuum state produced in type II down-conversion can be seen both as two copies of approximate EPR states, and also as a kind of polarization supersinglet, we show a straightforward way to test violations of the EPR concepts with direct use of their state. The observables involved are simply photon numbers at outputs of polarizing beam splitters. Suitable chained Bell inequalities are based on the geometric concept of distance. For a few settings they are potentially a new tool for quantum information applications, involving observables of a nondichotomic nature, and thus of higher informational capacity. In the limit of infinitely many settings we get a Greenberger-Horne-Zeilinger-type contradiction: EPR reasoning points to a correlation, while quantum prediction is an anticorrelation. Violations of the inequalities are fully resistant to multipair emissions in Bell experiments using parametric down-conversion sources.