We investigate the dynamics of entanglement between two continuous variable quantum systems. The model system consists of two atoms in a harmonic trap which are interacting by a simplified s-wave scattering. We show, that the dynamically created entanglement changes in a steplike manner. Moreover, we introduce local operators which allow us to violate a Bell-CHSH inequality adapted to the continuous variable case. The correlations show nonclassical behavior and almost reach the maximal quantum mechanical value. This is interesting since the states prepared by this interaction are very different from any EPR-like state.Comment: 9 page
We show that s waves, that is wave functions that only depend on a hyperradius, are entangled if and only if the corresponding Wigner functions exhibit negative domains. We illustrate this feature using a special class of s waves which allows us to perform the calculations analytically. This class includes a Gaussian, a maximally entangled as well as a "shell" state.
We discuss the estimation of channel parameters for a noisy quantum channel-the so-called Pauli channel-using finite resources. It turns out that prior entanglement considerably enhances the fidelity of the estimation when we compare it to an estimation scheme based on separable quantum states.
We present an operational definition of the Wigner function. Our method relies on the Fresnel transform of measured Rabi oscillations and applies to motional states of trapped atoms as well as to field states in cavities. We illustrate this technique using data from recent experiments in ion traps [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796] and in cavity QED [B. Varcoe et al., Nature 403, 743 (2000)]. The values of the Wigner functions of the underlying states at the origin of phase space are W |0 (0) = +1.75 for the vibrational ground state and W |1 (0) = −1.4 for the one-photon number state. We generalize this method to wave packets in arbitrary potentials.Currently the Wigner function [1] enjoys a renaissance in many branches of physics, ranging from quantum optics [2], nuclear [3] and solid state physics to quantum chaos [4]. This renewed interest has triggered a search for operational definitions [5] of the Wigner function, that is, definitions which are based on experimental setups [6]. In the present paper we propose such a definition based on the resonant interaction of an atom with a single mode of the electromagnetic field. In contrast to earlier work it is the atomic dynamics that performs the major part of the reconstruction. Our definition is well-suited for the Jaynes-Cummings model [7] but allows a generalization to other quantum systems.Many methods to reconstruct the Wigner function [7] of a cavity field or the motional state of a harmonic oscillator have been proposed [8]. In particular, the Wigner function of a cavity field can be expressed in terms of the measured atomic inversion [9]. This operational scheme [10] lives off the dispersive interaction between the atom and the field. Since this scheme requires long interaction times, an operational definition of the Wigner function based on a resonant interaction is desirable. The method of nonlinear homodyning [11] and quantum state endoscopy [12] fulfill this need, but require rather complicated reconstruction schemes.In contrast, the Fresnel representation proposed in the present paper is rather elementary. It expresses the value of the Wigner function at a phase space point α as a weighted time integral of the measured atomic dynamics caused by the state displaced by α. The weight function is the Fresnel phase factor. Since this method can be applied to a cavity field as well as a trapped ion [13] we use a harmonic oscillator as a model system. However, we show that this approach can easily be generalized to non-harmonic oscillators.Our definition relies on controlled displacements [13,14] of the quantum state of interest and the observation of Rabi oscillations of a two-level atom interacting resonantly with this field. We record the probability [15]of finding the atom in the ground state |g as a function of dimensionless interaction time τ and complex-valued displacement α. Here P n (α) ≡ n|D(α)ρD † (α)|n denotes the occupation statistics of the stateρ displaced by the displacement operatorD(α).The Wigner function W (α) of the orig...
We draw attention to various aspects of number theory emerging in the time evolution of elementary quantum systems with quadratic phases. Such model systems can be realized in actual experiments. Our analysis paves the way to a new, promising and effective method to factorize numbers. IntroductionInterference of waves instead of Newton's light corpuscles --what a revolutionary idea of Thomas Young pronounced in his famous talk at the Royal Society in 1801 [1]. The opposite idea --particles as waves --proposed by Louis de Broglie and cast into the appropriate mathematical form by Erwin Schrö dinger more than 100 years later opened a new realm in atomic physics with more applications today than ever. ''Can a quantum-mechanical description of physical reality be considered complete?" This famous question raised by Albert Einstein, Boris Podolsky and Nathan Rosen in their seminal paper [2] brought out most clearly the importance of entanglement [3] as the basic ingredient of quantum theory. Interference, wave mechanics and entanglement are the three ideas on which Peter Shor could build his factorization algorithm [4] with a polynomial rather than exponential speed. This discovery gave birth to the new field of quantum information [5]. Guided by the same three central ideas we show in the present paper that the time evolution of elementary quantum systems carries intrinsically the tools to factorize numbers [6].Stationary states have played an important role in the development of quantum mechanics. For this reason the time evolution of wave packets has for many years taken a back seat even in textbooks on quantum theory. In this context it is interesting to recall that already in 1927 Kennard [7] studied the time evolution of wave packets in simple potentials [8]. Recently, sophisticated techniques to create short laser pulses have opened a new era for a controlled creation and observation of wave packets in atoms [9], molecules [10] and cavities [11]. Even wave packet dynamics of cold atoms [12] and Bose-Einstein condensates [13] has been observed. These examples illustrate convincingly that today's technology allows to promote this gedankenexperiment of factorization to a real implementation in a quantum system.The use of wave packet dynamics to factorize numbers has been recognized for special examples. Clauser and Dowling [14] uses a N-slit Young interference setup. Hence, the number to be factorized is encoded in the number of slits. The indicator of a factor is the interference pattern on a screen as a function of the wavelength of the light or matter wave. Whenever the intensity across the screen is constant, the appropriately scaled wavelength provides a factor. Similarly, Harter [15] considers the time evolution of a particle in a box. In contrast to these contributions we provide a general framework for factorization and moreover, exhaust the potential of interference by capitalizing on properties of quadratic phase factors.
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