Complementarity and uncertainty relations for matter-wave interferometry

We establish a rigorous quantitative connection between (i) the interferometric duality relation for which-way information and fringe visibility and (ii) Heisenberg’s uncertainty relation for position and modular momentum. We apply our theory to atom interferometry, wherein spontaneously emitted photons provide which-way information and unambiguously resolve the challenge posed by the metamaterial “perfect lens” to complementarity and to the Heisenberg-Bohr interpretation of the Heisenberg microscope thought experiment.

Figure 1

Model for the Heisenberg microscope. The two atomic wave functions are located close to the origin and are separated by a small distance $a$ in the $x$ direction. The spontaneously emitted light that is collected in the detector propagates paraxially along the $z$ axis. The lens is located at $z=2f$ and the detector at $z=4f$.

Figure 2

(Color online) Verification of the duality relation for a set of 1000 random states. The duality relation is valid for states inside the orange circle; see text for details.

Figure 3

A combination of a wave packet (solid line) and a detector function (dashed line) for which there is no relation between which-way information and position uncertainty. The dotted line corresponds to ${\mid \mathcal{D}\mid }^2$.

Figure 4

(Color online) Which-way information (dashed line), visibility (dotted line), and duality $W^2+V^2$ (solid line) for a thin lens detector and well-separated wave packets as a function of the separation $a$ of the wave packets in units of the detector resolution $w_{\mathrm{eff}}$.

Figure 5

(Color online) Modulus (solid line) and phase (dashed line) of the mean modular momentum for a thin lens detector and well-separated wave packets as a function of the separation $a$ of the wave packets in units of the detector resolution $w_{\mathrm{eff}}$.