Abstract
We provide a theoretical framework to describe the interaction of a propagating guided matter wave with a localized potential in terms of quantum scattering in a confined environment. We analyze how this scattering correlates the longitudinal and transverse degrees of freedom, and work out analytically the output state under the Born approximation using a Gaussian localized potential. In this limit, it is possible to engineer the potential and achieve coherent control of the output channels. The robustness of this approximation is studied by comparing the stationary scattering theory to numerical simulations involving incident wave packets. It remains valid in a domain of weak localized potential that is achievable experimentally. We deduce a possible method to determine the longitudinal coherence length of a guided atom laser. Then, we detail the non-perturbative regime of the interaction of the guided matter wave with the localized potential using a coupled channel approach. This approach is worked out explicitly with a square potential. It yields new non-perturbative effects such as the occurrence of confinement-induced resonances. The perspectives opened up by this work for experiments are discussed.
Highlights
We show how the coupling between external degrees of freedom that occurs in the localized potential region generates controlled entangled states
The calculations described in this paper are motivated by the recent realization of guided atom lasers in the ground state of the transverse confinement [1]–[4]
The interaction of the matter wave with a localized potential, where open and closed channels are accessible through the scattering of the matter wave on the potential, is in direct analogy with non-reactive chemical reaction and guided electronic wave propagation in mesoscopic physics
Summary
To study the interaction of the guided atom laser with the defect, we determine the scattering states by solving the stationary Schrödinger equation,. Is the retarded propagator, 1 is the identity matrix and |φ0 is a solution in the absence of the defect H0|φ0 = E|φ0 This formulation is well suited to a formal perturbative expansion in powers of the localized potential U ,. For an incident wave function of the form |k0, 0 , the interaction of the atom laser with the localized potential produces the contamination of the modes |kn, n in the forward direction and. From equations (7) and (8) we find that the output wave resulting from the interaction of the incident wave with the defect is an entangled state, that is, a linear superposition of correlated bipartite states involving both a transverse state of quantum number n and a specific longitudinal state ±kn. For a sufficiently long propagation time, one expects the packet to split into a sum of packets if the initial dispersion δk in k is small enough (hkδk mω⊥)
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