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Entanglement and Wave Packet Structures in Photoionization and Other De- cay Processes of Bipartite Systems

Mikhail Vladimirovich Fedorov

Published Mar 27, 2009• 0 citations

Abstract

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.

DOI: 10.1201/9781584887799-19

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We investigate spectral and temporal features and entanglement of biphoton wave packets formed in spontaneous parametric down-conversion with a pulsed laser pump. The degree of entanglement is characterized by the experimentally measurable parameter

R

defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration $\ensuremath{\tau}$. The function $R(\ensuremath{\tau})$ is shown to have a minimum and even in the minimum, at rather natural conditions, the degree of entanglement is found to be very high

({R}_{\mathrm{min}}=73)

. The Schmidt number

K

is found analytically for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of

R

and

K

are found to be identical and numerical coefficients are found to be rather close. Two-time temporal wave function of a biphoton state is investigated in detail, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter

{R}_{t}

(defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number

K

.

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.