Single-photon single-ion interaction in front of a parabolic mirror Magdalena Stobińska, Robert Alicki, Gerd Leuchs Erlangen-Nürnberg University Max Planck Institute for the Science of Light 11th International Conference on Squeezed States and Uncertainty Relations 4th Feynman Festival June 22–26, 2009, Olomouc, CZECH REPUBLIC

Motivation for our Work Quantum Information Processing ● single - atom single - photon interface ● quantum teleportation ● atom chips Scalable Quantum Computation ● distributed quantum networks ● quantum repeater ● quantum memory Fundamental Process Study Spontaneous Emission ICSSUR' 09 Olomouc

Intuition: Spontaneous Emission What if we time reversed the spontaneous emission? ICSSUR' 09 Olomouc

Intuition: the Vectorial Properties of Light ● Strongly focused light improves coupling S. Quabis, et al, Opt. Commun. 179, 1 (2000) ● Increasing the overlap of light and the dipole wave improves focusing S. J. van Enk, PRA 69, (2004) ● Tailoring the polarisation pattern helps focusing R. Dorn, S. Quabis and G. Leuchs, PRL 69, (2003) S. J. van Enk and H. J. Kimble, PRA 63, (2001) ICSSUR' 09 Olomouc

Our Setup: Cavity or Free Space? Single photon is in a radially polarized donut mode Mirror ensures 4 π angle at the focus where the atom is located Free space configuration ICSSUR' 09 Olomouc

Cavity ● Small cavity Jaynes-Cummings model: interaction with one radiation mode collapses and revivals R. J. Cook and P. W. Milonni, PRA 35, 5081 (1987). P. Goy et al, PRL 50, 1903 (1983). ICSSUR' 09 Olomouc Decay rate quality factor free space value

Cavity or Free Space? ● Large cavity or half-cavity G. Alber, PRA 46, R5338 (1992). Wigner-Weisskopf model: interaction with many radiation mode damped collapses and revivals ICSSUR' 09 Olomouc

Atom-Light Interaction Model ● Evaluate spontaneous emission rate ● Dipole interaction atom dipole moment normal modes of quantized radiation field ICSSUR' 09 Olomouc

Normal Modes ● Helmholtz equation ● Transversality condition ICSSUR' 09 Olomouc

Normal Modes ICSSUR' 09 Olomouc

● Difficulties with transversality condition and boundary condition Problems with ● Boundary condition J. U. Nockel et al, PRE 62, 8677 (2000). mirror focal length ICSSUR' 09 Olomouc

● Modes for a given frequency Normal Modes in Free Space ICSSUR' 09 Olomouc C. P. Boyer et al, Nagoya Math. J. 60, 35 (1976).

- orthonormal set ● Transversality condition satisfied authomatically ● The modes are orthonormal ✔ quantum numbers ✔ polarization ✔ momentum Normal Modes in Free Space ICSSUR' 09 Olomouc

● Key assumption ● Hamiltonian Atom-Light Interaction atom is not driven by back reflected “rays” atom feels boundaries ICSSUR' 09 Olomouc

● Decay rate ● Free space Spontaneous Emission running waves ICSSUR' 09 Olomouc

● Asymptotic solution standing waves Normal Modes with Parabolic Mirror atom is far away form the mirror surface ● Boundary condition in exp: ICSSUR' 09 Olomouc

correction of order of Normal Modes with Parabolic Mirror ● Boundary condition ● Completeness relation in experiment ICSSUR' 09 Olomouc

● Standing waves in presence of the mirror ● Decay rate Spontaneous Emission atom position ICSSUR' 09 Olomouc

Spontaneous Emission ● Decay rate for the atom on the mirror axis ● Decay rate correction ICSSUR' 09 Olomouc

Tailored Resevouar ● Decay rate correction for small values of the wavevector ICSSUR' 09 Olomouc

Free Space ● Decay rate correction for large value of the wavevector ICSSUR' 09 Olomouc

Interference Effect ● Energy distribution depends on atom position ICSSUR' 09 Olomouc

Planned Experiment ● cooled ion as a two-level system ● and electronic levels as the ground and the excited state ● no hyperfine structure ● atomic transition frequency is in the ultraviolet regime ICSSUR' 09 Olomouc

Planned Experiment ● One photon pulse generation ● electro-optic modulation of a single-photon wave packet ● strongly attenuated laser pulse ● properly shaped single-photon Fock state wave function conditionally using photon pairs from PDC ICSSUR' 09 Olomouc P. Kolchin et al, PRL 101, (2008). N. Gisin et al, Rev. Mod. Phys. 74, 145 (2002).

Thank You! arXiv: ICSSUR' 09 Olomouc EPL 86, 14007, (2009) In discussion with: Gernot Alber Luis Sanchez-Soto