Arrival times in quantum mechanics by Seidel D.

By Seidel D.

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The momentum transfer from the spontaneously emitted photon onto the atom is included in Eq. 41), which represents the contrast to the case of an atom at rest. 44) −∞ (2) d3 x |ψt (x)|2 , where Parseval’s theorem has been used in Eq. 45) dΩk 1 − |ek ·D21 |2 |D21 |2 = dϑ (1 − cos2 ϑ) = 8π/3. The reset state for one-dimensional atomic motion can be obtained from Eq. 41) by taking the trace over the momentum components py and pz [83]. 5 Chapter 3. 32) the longitudinal Doppler effect, an effect of the order of v/c, is included in the present approach.

It is equivalent to the Monte-Carlo wave-function approach [88] and to the method of quantum trajectories [89]. A detailed review can be found by Plenio and Knight [90]. In this section the basic idea of the quantum jump approach is explained and the approach is used to describe the time evolution of a single two-level atom with quantized center-of-mass (cm) motion in the presence of spatially confined laser fields. This will provide the technical basis for the quantum optical arrival-time model of this work.

66), and the probability for the observation of the first photon at t is given by Eq. 70). The difference here is the probability of no emission at all. Since the wave packet for large t contains contributions from the reflected and from the transmitted part, it is asymptotically given by ∞ dk ψ(k) e−i Ψ(x, t) ∼ k 2 t/2m 0 R1 (k)e−ikx Θ(−x) + T1 (k)eikx Θ(x) |1 , t → ∞. 123) With Eq. 67) it follows that ∞ N (∞) = 0 dk |R1 (k)|2 + |T1 (k)|2 |ψ(k)|2 . 124) This expression never vanishes, since either the reflection or the transmission probability of the ground state is larger than zero.

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