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arXiv:1107.0636 · Journal of Physics B Atomic Molecular and Optical Physics

The Bragg regime of the two-particle Kapitza–Dirac effect

Pedro Sancho

Published Jun 29, 2011• 7 citations
arXivHTMLPDF

Abstract

We analyse the Bragg regime of the two-particle Kapitza–Dirac arrangement, completing the basic theory of this effect. We provide a detailed evaluation of the detection probabilities for multi-mode states, showing that a complete description must include the interaction time in addition to the usual dimensionless parameter w . The arrangement can be used as a massive two-particle beam splitter. In this respect, we present a comparison with Hong–Ou–Mandel-type experiments in quantum optics. The analysis reveals the presence of dips for massive bosons and a differentiated behaviour of distinguishable and identical particles in an unexplored scenario. We suggest that the arrangement can provide the basis for symmetrization verification schemes.

DOI
10.1088/0953-4075/44/14/145002

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We analyse the Bragg regime of the two-particle Kapitza–Dirac arrangement, completing the basic theory of this effect. We provide a detailed evaluation of the detection probabilities for multi-mode states, showing that a complete description must include the interaction time in addition to the usual dimensionless parameter w . The arrangement can be used as a massive two-particle beam splitter. In this respect, we present a comparison with Hong–Ou–Mandel-type experiments in quantum optics. The analysis reveals the presence of dips for massive bosons and a differentiated behaviour of distinguishable and identical particles in an unexplored scenario. We suggest that the arrangement can provide the basis for symmetrization verification schemes.
The celebrated Hong-Ou-Mandel effect is the paradigm of two-particle quantum interference. It has its roots in the symmetry of identical quantum particles, as dictated by the Pauli principle. Two identical bosons impinging on a beam splitter (of transmittance 1/2) cannot be detected in coincidence at both output ports, as confirmed in numerous experiments with light or even matter. Here, we establish that partial time reversal transforms the beam splitter linear coupling into amplification. We infer from this duality the existence of an unsuspected two-boson interferometric effect in a quantum amplifier (of gain 2) and identify the underlying mechanism as time-like indistinguishability. This fundamental mechanism is generic to any bosonic Bogoliubov transformation, so we anticipate wide implications in quantum physics.
Identical two-particle interferometry provides a scenario where interference and exchange effects manifest at once. We present a detailed calculation of the detection patterns in the two-particle two-slit experiment by extending Feynman's Gaussian slit approach to multi-mode states. We show the existence of two regimes depending on the mode distributions. In one of them we find a novel behavior for bosons, with detection patterns almost equal to those of distinguishable particles although the overlapping is large. As a byproduct of our analysis we conclude that the interaction at the diffraction grating is an efficient method to increase the initial overlapping. We propose a scheme, within the reach of present day technology, to experimentally verify our results.
Hong-Ou-Mandel (HOM) effect is known to be one of the main phenomena in quantum optics. The effect occurs when two identical single-photon waves enter a 1:1 beam splitter, one in each input port. When the photons are identical, they will extinguish each other. One of the main elements of the HOM interferometer is the beam splitter, which has its own coefficients of reflection R=1/2R = 1/2R=1/2 and transmission $ T = 1/2 $. In this work, the general mechanism of the interaction of two photons in a beam splitter is considered using an analytical solution, which shows that in the HOM theory of the effect it is necessary to know (including when planning the experiment) not only $ R = 1/2 $ and $ T = 1/2 $, but also their root-mean-square fluctuations $ ΔR ^ 2, ΔT ^ 2 $, which arise due to the dependence of $R = R(ω_1, ω_2) $ and $ T = T (ω_1, ω_2) $ on the frequencies where ω1,ω2ω_1, ω_2ω1​,ω2​ are the frequencies of the first and second photons, respectively. Under certain conditions, specifically when the dependence of the fluctuations $ ΔR^2 $ and $ ΔT^2 $ can be neglected and $ R=T=1/2 $ is chosen, the developed theory coincides with previously known results.
A relativistic description of the Kapitza-Dirac effect in the so-called Bragg regime with two and three interacting photons is presented by investigating both numerical and perturbative solutions of the Dirac equation in momentum space. We demonstrate that spin flips can be observed in the two-photon and the three-photon Kapitza-Dirac effects for certain parameters. During the interaction with the laser field the electron's spin is rotated, and we give explicit expressions for the rotation axis and the rotation angle. The off-resonant Kapitza-Dirac effect, that is, when the Bragg condition is not exactly fulfilled, is described by a generalized Rabi theory. We also analyze the in-field quantum dynamics as obtained from the numerical solution of the Dirac equation.
The Hong–Ou–Mandel (HOM) dip has played an important role in recent linear optics experiments. It is crucial for quantum computing with photons and can be used to characterize the quality of single photon sources and linear optics setups. In this paper, we consider generalized HOM experiments with N bosons or fermions passing simultaneously, i.e. within their coherence time, through a symmetric Bell multiport beam splitter. It is shown that for an even number of bosons, the HOM dip occurs naturally in the coincidence detection in the output ports. In contrast, fermions always leave the setup separately, exhibiting perfect coincidence detection. Our results can be used to verify or employ the quantum statistics of particles experimentally.