Quantum interference in a driven washboard potential

F.

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H. G.

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M. Ben

Dahan

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C. F.

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M. G.

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Q.

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X. G.

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G. A.

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M. G.

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M.

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J. H.

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M. G.

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English translation: Photon and Atoms: Basic Processes and Applications (Wiley, New York, 1997).

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Several simple Maple programs for visualization of the results are available from 〈http://www.phlam.univ-lille1.fr/atfr/cq〉.

N. W. Ashkroft and N. D. Mermin, Solid State Physics (Saunders College, Fort Worth, TX, 1976).

We suppose that the system is enclosed in a large bounding box (containing many wells $L≫1000)$ and consider the bulk properties that are not altered by boundary effects.

By virtue of the translational properties of the WS states, Eq. (8), $Xn,n+p$ $(p≠0)$ does not depend on $n:$ $Xn,n+p=∫φn*(x)xφn+p(x)dx=∫φ0*(x−nd)xφp(x−nd)dx=∫φ0*(x′)(x′+nd)φp(x′)dx′=X0,p$ where we used the orthogonality of the WS states. Note, however, that $Xn,n=X0,0+n$ does depend on $n.$

It is important to note that WS states are localized in a well, but not perfectly: their wave function present a small secondary maximum in the neighbor wells. This secondary maximum decreases as the depth of the well increases. As a consequence, the superposition integrals of the type $Xp=∫φn(x)xφn+p(x)$ are not zero for $p∼1,$ but rapidly decrease with $p.$ For $F=3$ and $V0=25,$ we obtain numerically $X1=0.13$ (coupling to the next-neighbor, generating oscillations at frequency $ωB)$⁠, $X2=7.8×10−3$ (coupling to next-to-next neighbor, generating oscillations at frequency $2ωB)$⁠. The amplitude of the oscillation at $2ωB$ is thus roughly 20 times smaller than that at $ωB.$

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This approximation is not necessary, but it simplifies the calculations without any loss of physical effects. For a treatment without this approximation, see Q. Thommen, J. C. Garreau, and V. Zehnlé, “Atomic motion in tilted optical lattices,” Arxiv quant-ph/0307118 (2003).

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E. Butkov, Mathematical Physics (Addison-Wesley, Reading, MA, 1973), for example.

Q.

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,

J. C.

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, and

V.

Zehnlé

, “,” , – ().