Publication: Exact energy-time uncertainty relation for arrival time by absorption

Jukka Kiukas1, Andreas Ruschhaupt1, Piet O Schmidt2,3 and Reinhard F Werner1

1 Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr 2, 30167 Hannover, Germany
2 QUEST Institute for Experimental Quantum Metrology, Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
3 Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wavefunction, we show that $\Delta T\Delta E\ge \sqrt{p}\hbar /2$ and $\langle T\rangle \Delta E\ge 1.37\sqrt{p}\hbar$, where 〈T〉 denotes the mean arrival time and p is the probability for the particle to be eventually absorbed. Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.

Reference: Jukka Kiukas et al 2012 J. Phys. A: Math. Theor.45 185301