Author

# Friedrich Wilhelm Schäfke

Bio: Friedrich Wilhelm Schäfke is an academic researcher from University of Mainz. The author has contributed to research in topics: Mathieu function. The author has an hindex of 7, co-authored 13 publications receiving 807 citations.

Topics: Mathieu function

##### Papers

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01 Jan 1980

103 citations

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01 Jan 1980

90 citations

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01 Jan 1954

84 citations

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23 citations

##### Cited by

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TL;DR: In this paper, the transmon was proposed to operate in a regime of significantly increased ratio of Josephson energy and charging energy, while maintaining sufficient anharmonicity for selective qubit control.

Abstract: Short dephasing times pose one of the main challenges in realizing a quantum computer. Different approaches have been devised to cure this problem for superconducting qubits, a prime example being the operation of such devices at optimal working points, so-called ``sweet spots.'' This latter approach led to significant improvement of ${T}_{2}$ times in Cooper pair box qubits [D. Vion et al., Science 296, 886 (2002)]. Here, we introduce a new type of superconducting qubit called the ``transmon.'' Unlike the charge qubit, the transmon is designed to operate in a regime of significantly increased ratio of Josephson energy and charging energy ${E}_{J}∕{E}_{C}$. The transmon benefits from the fact that its charge dispersion decreases exponentially with ${E}_{J}∕{E}_{C}$, while its loss in anharmonicity is described by a weak power law. As a result, we predict a drastic reduction in sensitivity to charge noise relative to the Cooper pair box and an increase in the qubit-photon coupling, while maintaining sufficient anharmonicity for selective qubit control. Our detailed analysis of the full system shows that this gain is not compromised by increased noise in other known channels.

2,807 citations

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TL;DR: In this paper, the authors apply the theory developed in the preceding paper to a number of questions about timelimited and bandlimited signals, and find the signals which do the best job of simultaneous time and frequency concentration.

Abstract: The theory developed in the preceding paper1 is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.

2,498 citations

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TL;DR: The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.

Abstract: Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.

1,569 citations

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01 Dec 1963

TL;DR: In this article, the first order statistics of the observed electric-field strength, the observed light intensity, and observed light phase are examined, and the autocorrelation functions of the complex field and intensity processes are investigated, and that of the electric field is found to be proportional to the Fourier transform of the light intensity distribution incident on the scattering surface.

Abstract: : When laser light strikes a diffuse object, such as paper, the scattered light has been observed to possess a granular spatial structure. The statistical properties of these so-called 'sparkle patterns,' as seen by an observer in the far field of the scattering spot, are investigated. The first order statistics of the observed electric-field strength, the observed light intensity, and the observed light phase are examined. The electric field is reasoned to be a complex normal random variable; the intensity a real, exponentially distributed random variable; and the phase a uniformly distributed random variable. Higher order statistics of these random processes are also discussed. The autocorrelation functions of the complex field and the intensity processes are investigated, and that of the electric field is found to be proportional to the Fourier transform of the light-intensity distribution incident on the scattering surface. Spatial averages of the light intensity are considered and are found to converge to corresponding ensemble averages when either the area of the scattering spot or the average area grows large.

1,526 citations

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01 Jan 1975

TL;DR: In this article, the first-order statistics of the complex amplitude, intensity and phase of speckle are derived for a free-space propagation geometry and for an imaging geometry.

Abstract: Since speckle plays an important role in many physical phenomena, it is essential to fully understand its statistical properties. Starting from the basic idea of a random walk in the complex plane, we derive the first-order statistics of the complex amplitude, intensity and phase of speckle. Sums of speckle patterns are also considered, the addition being either on an amplitude or on an intensity basis, with partially polarized speckle being a special case. Next we consider the sum of a speckle pattern and a coherent background, deriving the first-order probability density functions of intensity and phase. Attention is then turned to second-order statistics. The autocorrelation function and power spectral density are derived, both for a free-space propagation geometry and for an imaging geometry. In some cases the recorded speckle pattern may be spatially integrated or blurred, and accordingly consideration is given to the statistics of such patterns. Finally, the relationship between detailed surface structure and the resulting speckle pattern is explored, with emphasis on the effects of the surface autocorrelation function and the effects of finite surface roughness.

1,217 citations