Quantum Imaging with Trapped Ions

1. Quantum Imaging with Trapped Ions Joachim von Zanthier1 Christoph Thiel1 Thierry Bastin2 Enrique Solano3 Girish S. Agarwal4 1Institut für Optik, Information und Photonik, Universität Erlangen-Nürnberg, Germany 2Institut de Physique Nucléaire, Atomique et de Spectroscopie, Université de Liège au Sart Tilman, Belgium 3Departamento de Química Física, Universidad del País Vasco - Euskal Herriko Unibertsitatea, Bilbao, Spain 4Department of Physics, Oklahoma State University, Stillwater, USA

2. Quantum Imaging with Trapped Ions Content: I. First and Second Order Correlations in the Fluorescence Light of trapped Ions II. Using higher Order Correlation Functions for Quantum Imaging III. Quantum Imaging of a photon source and an aperture

3. Young’s Double Slit Experiment Intensity distribution: constructive interference: First-order correlation function

4. laserfield interference fringes kL I. First-order correlations in the fluorescence light of trapped ions two trapped ions U. Eichmann et al., Phys. Rev. Lett. 70, 2359 (1993)

5. two 2-level atoms: uncorrelated atoms: W. M. Itano et al., PRA 57, 4176 (1998); C. Skornia et al. PRA 64, 063801 (2001) V= first-order correlation function:

6. coherently scattered light scattered intensity incoherently scattered light 0 2 4 6 8 10 12 sat. parameter s s 0 Mollow Triplett -4 -2 0 2 4 (w – w0)/G

7. pulsed regime: both ions initially in the excited state G(1) 1- no interferences 0- Detector position r complete „which-way-information“ available or

8. or I. Second-order correlations in the fluorescence light of trapped ions no which-way-information available regarding two possible quantum paths

9. Quantum Imaging? Introducing (Glauber’s) correlation functions R. J. Glauber, Phys. Rev. 130, 2529; 131, 2766 (1963) Spatial (intensity) correlation functions: Roy J. Glauber … for his contribution to the quantum theory of optical coherence

10. collective excitation N single photon detectors … … correlation measurement … … in far-fieldregion of the ions II. Using higher order correlation functions for Quantum Imaging |e N trapped2-level ions |g

11. Far-field Laser pulse Intensity-intensity correlation G(2) R2 d R1 Trap For two 2-level ions: and Example: Second order correlation function in the fluorescence light of two trapped ions

12. -1 -0 Example: Second order correlation function in the fluorescence light of two trapped ions

13. G(2) G(2) G(2) 1- 1- 1- r2 = r1 r2 = const. r2 = -r1 0- 0- 0- Detector position Detector position Detector position Example: Second order correlation function in the fluorescence light of two trapped ions -1 -0 G. S. Agarwal et al., PRA 70, 063816 (2004)

14. Generalisation to N th order correlations: RN … R3 Higher order correlations in the fluorescence light of trapped ions Far-field Laser pulse … N-times Intensity correlation G(N) Intensity-intensity correlation G(2) R2 d R1 Trap

15. for n = N : where Example: 4th order correlation function for 4 ions and 4 detectors (N = 4) for any detector positions d1, d2, d3, d4 :

16. choose : choose : choose :

17. G(2) Detector position G(2) G(4) Detector position Detector position Higher order correlations in the fluorescence light of trapped ions G(2) r2 = r1 Detector position r2 = const. r2 = -r1

18. One can show: for any n = N, one can always find suitable detector positions so that one obtains a singlesinusoidal modulation with wave number: C. Thiel, T. Bastin, J. Martin, E. Solano, J. von Zanthier, G. S. Agarwal, PRL 99, 133603 (2007)

19. +2. Abbe-Limit: +1. d·sinq=± l q d 0. d ≥ -1. -2. object plane Fourier plane image plane l only if: NA Fringe spacing = Measure of resolution Resolution in classical optics Abbe’s theory of the microscope:“to construct an image from an object the first diffraction order in the Fourier plane must be at least visible”

20. -p/2 0 +p/2 -p/2 0 +p/2 III. Application: quantum imaging of 8 trapped ions example: 8 ions with separation 2·l detected with versus :

21. Spacing ofthe ions: Classicmicroscopy Quantummicroscopy Abbe’s theory of microscopic resolution:“to construct an image from an object the first diffraction order in the Fourier plane must be at least visible” III. Application: quantum imaging of 8 trapped ions example: 8 ions detected with versus :

22. G(2) G(2) III. Application: quantum imaging of an object Light from two atoms passing through an aperture Ch. Thiel et al., quant-ph/0805.1831

23. G2 for two atoms and two detectors with resolution increased by “2”: G(2) Classical result with one detector: Fresnel-Kirchhoff diffraction formula: for: and: Ch. Thiel et al., quant-ph/0805.1831 III. Application: quantum imaging of an object classical and quantum imaging of a rectangular aperture • in analogy to classical diffraction theory: • Fresnel-Kirchhoff diffraction • Fraunhofer approximation • paraxial approximation

24. Quantum imaging of an aperture with G(2) using a lens G(2) for r2 = r1 1 0 detector position lens III. Application: quantum imaging of an object Generalization of the quantum imaging scheme Quantum imaging of an aperture with G(2) G(2) for r2 = - r1 1 r1 0 r2 detector position r2=r1

25. Nth-order correlation functionscan be used for imaging: quantum imaging of a source with visibility of 100% and resolution of l/N Conclusion - Higher order correlation functions show interference pattern even if measured with (in 1. order) incoherent light - Interference pattern of second order correlation function displays visibility of 100% and resolution enhanced by a factor of ”2“ - Same scheme can be used for quantum imaging of an object, either in the Fourier plane of the object or in the image plane of a lens - Same scheme can be used for generating entangled states in long-living internal levels of the photon emitting atoms (see poster session today and Thursday)

26. ”Thank you“