1. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN Characterization and manipulation of frequency entangled qudits 1 Bänz Bessire Quantum Optics Lab – The Stefanov Group Institute of Applied Physics, University of Bern, Switzerland Quantum 2014, Turin, 27.05.2014

2. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 27.05.2014Bänz Bessire2 13 Introduction The dit is a -dimensional extension of a classical 2-level system (bit) From a bit to a dit From a qubit to a qudit The qudit is a -dimensional extension of a quantum 2-level system (qubit) … … … … … …

3. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire3 13 Qudit states in quantum information Quantum key distribution L. Sheridan and V. Scarani, Phys. Rev. A 82, 030301 (2010) Increase secret bit key rate Increase robustness to noise dI d (maximally entangled state) I d (non-maximally entangled state) 22.82 32.872.91 42.903.02 52.913.05 Lower detection efficiencies required for detection loophole free Bell tests T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010) 27.05.2014 Fundamental tests of quantum mechanics Higher robustness to noise in Bell measurements Larger violation of Bell inequalities D. Collins et al., Phys. Rev. Lett. 88, 032322 (2013) A. Acín, T. Durt, N. Gisin, and J. I. Latorre, Phys. Rev. A 65, 052325 (2002)

4. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN Polarization entanglement (qubits) P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337–4342 (1995) many more Transverse momentum entanglement (qudits) A. Mair et al., Nature 412, 313-6 (2001) A. Dada et al., Nature Physics 7, 677-680 (2011) M. Agnew et al., Phys. Rev. A 84, 062101 (2011) 1 Bänz Bessire4 13 Photonic entangled qudit states Energy-time (frequency) entanglement (continuous) Discrete time bins (qudits): R. Thew et al., Phys. Rev. Lett. 93, 1–4 (2004) D. Richart et al., Appl. Phys. B 106, 543–550 (2012) Spontaneous parametric down-conversion (SPDC) (idler) (signal) (pump) 27.05.2014

5. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN Preparation (SPDC) Manipulation PPKTP Beam dump Detection (SFG) SPCM 5 W pump laser 1064 nm 700 s -1 up-converted photons 4-prism compressor: Compensates for dispersion Aligns the spectrum Spatial light modulator (SLM) SLM: Jenoptik S640d 532 nm PPKTP Bandpass filter Complex transfer function: 1 Bänz Bessire5 13 Experiment Setup Shaping of the two-photon wavefunction by spectral amplitude and phase modulation with Detected signal after manipulation (SLM) and SFG: A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, 073601 (2005) F. Zäh, M. Halder, and T. Feurer, Optics Express 16, 16452-16458 (2008) 27.05.2014

6. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire6 13 Frequency-bin entangled qudits Frequency bins Discretize the SPDC spectrum to obtain entangled qudits Imply a frequency-bin structure with the SLM Independent control Subdivide the spectrum into frequency bins according to 27.05.2014

7. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire7 13 Quantum state tomography C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, Phys. Rev. A 88, 032322 (2013) Reconstruction of the density matrices for maximally entangled qudits by Maximum Likelihood Estimation Frequency-bin entangled qudits Detected signal is equivalent to the signal of a projective measurement with SLM 27.05.2014

8. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN By means of projective measurements we determine 1 Bänz Bessire8 13 Frequency-bin entangled qudits 2-qubit / 2-qutrit state with a variable degree of entanglement Collins et al. (CGLMP) introduced a -dimensional Bell parameter Correlations between two separated systems are explainable by a local realistic theory if (CGLMP inequality) Bell measurements D. Collins et al., Phys. Rev. Lett. 88, 040404 (2002) is varied by amplitude modulation with the SLM 27.05.2014

9. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire9 13 Frequency-bin entangled qudits Bell measurements 2-qubit state with a : 2-qutrit state with a : C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, Phys. Rev. A 88, 032322 (2013) Theory scaled by mixing parameters acc. to A non-maximally entangled qutrit state ( ) violates the CGLMP inequality stronger than a maximally entangled qutrit A. Acín, T. Durt, N. Gisin, and J. I. Latorre, Phys. Rev. A 65, 052325 (2002) 27.05.2014

10. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire10 13 Time-bin entangled qudits Time bins with a SLM Analogue to frequency bins we subdivide the time domain into bins Using two time bins corresponds to the Franson concept of 2-photon interferometry SPDC J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989) FT 27.05.2014

11. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire11 13 Time-bin entangled qudits Demonstrate entangled qubits by 2-photon interference Projection of a 2-qubit state onto Finite spectral resolution at SLM plane reduces the visibility Coherence times: SLM B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

12. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire12 13 Outlook Implementation of other discretization schemes Discretization of the frequency space into Schmidt modes Non-locality measures Use Bell measurement data (qutrit) to calculate the non-local capacity and the distance to the non-local polytope as non-locality measures C. Bernhard, B. Bessire, A. Montina, M. Pfaffhauser, A. Stefanov, and S. Wolf, arXiv:1402.5026 (2014) Improvements of the experimental setup Enhance the detection efficiency by new detection schemes Improved optical resolution Allows for higher dimensions Demonstrate a Kochen-Specker (quantum game) experiment using entangled qutrits (or ququarts) Future projects Two-photon absorption with entangled photons (coherent control) B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

13. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire13 Acknowledgments Thank you for your attention! C. Bernhard S. Lerch S. Schwarz M. Unternährer A. Stefanov T. P. Wihler S. Wolf and his group Collaborators: J. Kohn 27.05.2014

14. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire BACKUP SLIDES 27.05.2014

15. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Entanglement quantification 27.05.2014

16. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Entanglement quantification Entropy of entanglement Reduced density matrix Consider the state of the i(s) subsystem Entropy of entanglement: Corresponds to entanglement in a maximally entangled qudit with T. Wihler, B. Bessire, and A. Stefanov, arXiv:1209.2575 (2012) CW pump field large discretization scheme is required: Compute without diagonlization of : Given a set of experimental parameters: How strong is the entanglement between the two photons? 27.05.2014

17. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Entanglement quantification Entropy 27.05.2014

18. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Entanglement quantification Schmidt number 27.05.2014

19. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire SLM UC Single photon limit 27.05.2014

20. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Experiment Point spread function (PSF) Diffraction effects due to the finite aperture of lenses and other optical elements lead to a point-to-spot image Single spectral components are spatially broadened at the plane of the SLM 27.05.2014

21. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Experiment SLM Spectral phase shift in each display: display 2 display 1 Two liquid crystal displays Polarization dependent detection (SFG) + Independent spectral amplitude and phase manipulation = 27.05.2014

22. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire = UC-crystal SLM working principle 27.05.2014

23. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Theory of SFG coincidence detection First order perturbation theory allowed due to low SFG efficiency: Propagated signal-idler field operators: {,, SLM, O, …} K. A. O’Donnell and A. B. U’Ren Phys. Rev. Lett. 103, 123603 (2009) Coincidence state : NL crystal SPC SFG 27.05.2014

24. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Theory of SFG coincidence detection Properties of the SFG crystal are included Transferfunction of an arbitrary optical setup is included 27.05.2014

25. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Theory of SFG coincidence detection Are the properties of the SFG crystal correctly incorporated in ? 27.05.2014

26. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Equivalence Modified joint spectral amplitude 27.05.2014

27. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Single photon limit Corresponds to a mean spectral photon density of Maximal flux in which photon pairs are well distinguishable from each other Maximal flux Experimental power Below single photon limit B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, 43602 (2005) 27.05.2014

28. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Additional Bell measurement informations 27.05.2014

29. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Frequency-bin entangled qudits Bell measurement The Bell parameter for and reads CGLMP inequality violated by entangled states Single contributions to by projections 2-qubit/2-qutrit state with a variable degree of entanglement 27.05.2014

30. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Frequency-bin entangled qudits I 2 and I 3 theory Curves are scaled to experimental data with corresponding mixing parameter Symmetric noise model with mixing parameter : 27.05.2014

31. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Frequency-bin entangled qudits Non-locality quantification of experimental qutrit pairs Three measures of non-locality Bell parameter Distance to the local polytope Asymmetric non-local capacity 27.05.2014

32. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Schmidt mode qudits Frequency bin qudits two-photon interference 27.05.2014

33. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN Schmidt decomposition 1 Bänz Bessire Schmidt-mode entangled qudits represents a pure two-photon state Schmidt decomposition Efficient approach to discretize into an entangled qudit state Basis for qudits are orthonormal 27.05.2014

34. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Schmidt-mode entangled qudits SLM transfer function for the entangled qubit SLM transfer function for the entangled qutrit B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

35. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Schmidt-mode entangled qudits Entanglement CGLMP inequality violation Coincidence signal for data fitting B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

36. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Schmidt-mode entangled qudits 27.05.2014

37. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Frequency-bin entangled qudits CGLMP inequality violation Entanglement for all B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

38. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Time-bin qudits 27.05.2014

39. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN Transition from a separable to a maximally entangled qubit 1 Bänz Bessire Time-bin entangled qubits Generalized qubit state Transition modelled by coincidence rate are used as fitting parameters For it holds that 27.05.2014

40. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN Fitting parameters 1 Bänz Bessire Time-bin entangled qubits Generalized qubit state B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

41. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Time-bin entangled qubits Bell parameter I 2 B. Bessire, C. Bernhard, T. Feurer, and A. Stefanov, New. J. Phys., 033017 (2014) 27.05.2014

42. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Time-bin entangled qubits Single photon intensity versus coincidence signal Measure single photon intensity with a powermeter and a linear polarizer The single photon intensity is given by Separable qubitMaximally entangled qubit 27.05.2014

43. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Time-bin entangled qubits Single photon intensity versus coincidence signal 27.05.2014

44. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Coherent control 27.05.2014

45. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Coherent control Interferometric autocorrelation measurement Entangled two-photon state enters a Mach-Zehnder interferometer with a subsequent SFG process The coincidence signal is a coherent superposition of SLM transfer function 27.05.2014

46. QUANTUM OPTICS LAB IAP, UNIVERSITÄT BERN 1 Bänz Bessire Coherent control Interferometric autocorrelation measurement F. Zäh, M. Halder, and T. Feurer, Optics Express 16, 16452-16458 (2008) 27.05.2014