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Two-particle breakup and Einstein-Podolsky-Rosen entanglement

Kam Wai Clifford Chan

Published Jan 1, 2005• 0 citations

Abstract

The problem of two-particle breakup involving Einstein- Podolsky-Rosen entanglement is studied in this thesis. One or both of the fragments in the breakup can be massive. We have examined photoionization, photodissociation, spontaneous emission, Raman scattering and Rayleigh scattering, and identified a generic control parameter that determines the degree of entanglement in these physical systems. This control parameter allows one to find the conditions to achieve very high entanglement. Similarities among the systems are also found in terms of entanglement and wave packet localization, while subtle differences in the evolution of wave packets due to the nature of the participating particles are observed. With the help of a Gaussian model, a concrete relation between entanglement and experimentally measurable quantities is established. The model also explains the apparent entanglement loss in the evolution of wave packets in free space through the notion of phase entanglement, and relates the Heisenberg uncertainty relation and Einstein- Podolsky-Rosen conditional uncertainty to the Schmidt number. It is conjectured that the Schmidt number serves as the upper or lower bound to the uncertainty relations for a generic bipartite breakup.

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The problem of two-particle breakup involving Einstein- Podolsky-Rosen entanglement is studied in this thesis. One or both of the fragments in the breakup can be massive. We have examined photoionization, photodissociation, spontaneous emission, Raman scattering and Rayleigh scattering, and identified a generic control parameter that determines the degree of entanglement in these physical systems. This control parameter allows one to find the conditions to achieve very high entanglement. Similarities among the systems are also found in terms of entanglement and wave packet localization, while subtle differences in the evolution of wave packets due to the nature of the participating particles are observed. With the help of a Gaussian model, a concrete relation between entanglement and experimentally measurable quantities is established. The model also explains the apparent entanglement loss in the evolution of wave packets in free space through the notion of phase entanglement, and relates the Heisenberg uncertainty relation and Einstein- Podolsky-Rosen conditional uncertainty to the Schmidt number. It is conjectured that the Schmidt number serves as the upper or lower bound to the uncertainty relations for a generic bipartite breakup.

A structure of wave packets arising in the process of a bipartite system decay is investigated and its connection with entanglement of quantum states is discussed. As long as the eﬀect of wave packet spreading can be ignored, the ratio R of the single-particle to coincidence wave packet widths is shown to coincide with the degree of entanglement determined by the Schmidt correlation number K. Mathematically single-particle and coincidence schemes correspond to absolute and conditional probability densities. Coordinate and momentum wave-packets for each particle are compared. A series of reciprocity and identity relations between the corresponding wave packet widths is found to occur. Uncertainty relations following from these relations are found. The coordinate and momentum uncertainty products for each particle are shown to be related directly to the degree of entanglement K: the single-particle and coincidence uncertainty products are shown to be, correspondingly, ∼ K ≥ 1 and ∼ 1/K ≤ 1. A series of speciﬁc examples are considered.

The narrowing of electron and ion wave packets in the process of photoionization is investigated, with the electron-ion recoil taken fully into account. Packet localization of this type is directly related to entanglement in the joint quantum state of the electron and ion, and to Einstein-Podolsky-Rosen localization. Experimental observation of such packet-narrowing effects is suggested via coincidence registration by two detectors, with a fixed position of one and varying position of the other. A similar effect, typically with an enhanced degree of entanglement, is shown to occur in the case of photodissociation of molecules.

Certification and quantification of correlations for multipartite states of quantum systems appear to be a central task in quantum information theory. We give here a unitary quantum-mechanical perspective of both entanglement and Einstein-Podolsky-Rosen (EPR) steering of continuous-variable multimode states. This originates in the Heisenberg uncertainty relations for the canonical quadrature operators of the modes. Correlations of two-party

(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)

-mode states are examined by using the variances of a pair of suitable EPR-like observables. It turns out that the uncertainty sum of these nonlocal variables is bounded from below by local uncertainties and is strengthened differently for separable states and for each one-way unsteerable state. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability of

(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)

-mode states in both possible ways of steering. When the states and the performed measurements are Gaussian, then these conditions are precisely the previously known criteria of separability and one-way unsteerability.

We present a method to detect bipartite entanglement based on number-phase-like uncertainty relations in split spin ensembles. First, we derive an uncertainty relation that plays the role of a number-phase uncertainty for spin systems. It is important that the relation is given with well-defined and easily measurable quantities, and that it does not need assuming infinite dimensional systems. Based on this uncertainty relation, we show how to detect bipartite entanglement in an unpolarized Dicke state of many spin-1/2 particles. The particles are split into two subensembles, then collective angular momentum measurements are carried out locally on the two parts. First, we present a bipartite Einstein-Podolsky-Rosen (EPR) steering criterion. Then, we present an entanglement condition that can detect bipartite entanglement in such systems. We demonstrate the utility of the criteria by applying them to a recent experiment given in K. Lange et al. [Science 360, 416 (2018)] realizing a Dicke state in a Bose-Einstein condensate of cold atoms, in which the two subensembles were spatially separated from each other. Our methods also work well if split spin-squeezed states are considered. We show in a comprehensive way how to handle experimental imperfections, such as the nonzero particle number variance including the partition noise, and the fact that, while ideally BECs occupy a single spatial mode, in practice the population of other spatial modes cannot be fully suppressed.