2. Quantum information with photons and atoms: from tomography to error correction C. W. Ellenor, M. Mohseni, S.H. Myrskog, J.K. Fox, J. S. Lundeen, K. J. Resch, M. W. Mitchell, and Aephraim M. Steinberg Dept. of Physics, University of Toronto Title slide PQE 2003

3. U of T quantum optics & laser cooling group: PDF: Morgan Mitchell Optics: Kevin Resch (  Wien)Jeff Lundeen Chris Ellenor (  Korea)Masoud Mohseni Reza Mir(  Lidar) Atom Traps: Stefan MyrskogJalani Fox Ana JofreMirco Siercke Salvatore MaoneSamansa Maneshi TBA: Rob Adamson Theory friends: Daniel Lidar, Janos Bergou, John Sipe, Paul Brumer, Howard Wiseman Acknowledgments

4. OUTLINE Introduction: Photons and atoms are promising for QI. Need for real-world process characterisation and tailored error correction. No time to say more. Quantum process tomography on entangled photon pairs - E.g., quality control for Bell-state filters. - Input data for tailored Quantum Error Correction. An experimental application of decoherence-free subspaces in a quantum computation. Quantum state (and process?) tomography on center-of-mass states of atoms in optical lattices. Coming attractions…

5. Density matrices and superoperators

6. HWP QWP PBS Argon Ion Laser Beamsplitter "Black Box" 50/50 Detector B Detector A Two-photon Process Tomography Two waveplates per photon for state preparation Two waveplates per photon for state analysis SPDC source

7. Hong-Ou-Mandel Interference > 85% visibility for HH and VV polarizations HOM acts as a filter for the Bell state:   = (HV-VH)/√2 Goal: Use Quantum Process Tomography to find the superoperator which takes  in   out Characterize the action (and imperfections) of the Bell- State filter.

8. “Measuring” the superoperator } Output DM Input HH HV VV VH } } } etc. 16 analyzer settings 16 input states Coincidencences

9. “Measuring” the superoperator Input Output DM HH HV VV VH etc. Superoperator Input Output

10. “Measuring” the superoperator Input Output DM HH HV VV VH etc. Input Output Superoperator

11. Testing the superoperator LL = input state Predicted N photons = 297 ± 14

12. Testing the superoperator LL = input state Predicted Observed N photons = 297 ± 14 N photons = 314

13. So, How's Our Singlet State Filter? Observed    Bell singlet state:   = (HV-VH)/√2

14. Model of real-world beamsplitter 45° “unpolarized” 50/50 dielectric beamsplitter at 702 nm (CVI Laser) birefringent element + singlet-state filter + birefringent element Singlet filter AR coating multi-layer dielectric

15. Predicted Best Fit:  1 = 0.76 π  2 = 0.80 π Model beamsplitter predicitons Singlet filter

16. Observed Predicted Comparison to measured Superop Predicted

17. H H xx H y y f(x) We use a four-rail representation of our two physical qubits and encode the logical states 00, 01, 10 and 11 by a photon traveling down one of four optical rails numbered 1, 2, 3 and 4, respectively. 1 2 3 4 Photon number basis 1 st qubit 2 nd qubit Computational basis OracleAA Performing a quantum computation in a decoherence-free subspace The Deutsch-Jozsa algorithm:

18. Modified Deutsch-Jozsa Quantum Circuit y f(x) H xx H y H 00 01 e i  01 11 10 e i  10 But after oracle, only qubit 1 is needed for calculation. Encode this logical qubit in either DFS: (00,11) or (01,10). Error model and decoherence-free subspaces Consider a source of dephasing which acts symmetrically on states 01 and 10 (rails 2 and 3)…

19. 3 4 1 2 1 2 4 23 Experimental Setup Oracle Optional swap for choice of encoding Preparation Random Noise Mirror Waveplate Phase Shifter PBS Detector A B C D 3/4 4/3 DJ experimental setup 3

20. CBCCCBBB DFS Encoding Original encoding Constant function Balanced function C B DJ without noise -- raw data

21. CBCCCBBB DFS Encoding Original Encoding C B Constant function Balanced function DJ with noise-- results

22. Tomography in Optical Lattices Part I: measuring state populations in a lattice…

23. Houston, we have separation!

24. Wait… Shift… Measure ground state population Quantum state reconstruction Initial phase- space distribution p x p x =x=x x p  t Q(0,0) = P g (More recently: direct density-matrix reconstruction)

25. Quasi-Q (Pg versus shift) for a 2-state lattice with 80% in upper state.

26. Exp't:"W" or [P g -P e ](x,p)

27. W(x,p) for 80% excitation

28. Coming attractions A "two-photon switch": using quantum enhancement of two-photon nonlinearities for - Hardy's Paradox (and weak measurements) - Bell-state determination and quantum dense coding(?) Optimal state discrimination/filtering (w/ Bergou, Hillery) The quantum 3-box problem (and weak measurements) Process tomography in the optical lattice - applying tomography to tailored Q. error correction Welcher-Weg experiments (and weak measurements, w/ Wiseman) Coherent control in optical lattices (w/ Brumer) Exchange-effect enhancement of 2x1-photon absorption (w/ Sipe, after Franson) Tunneling-induced coherence in optical lattices Transient anomalous momentum distributions (w/ Muga) Probing tunneling atoms (and weak measurements) … et cetera

29. Schematic diagram of D-J interferometer 1234 1 2 3 4 12 3 4 Oracle 00 01 10 11 “Click” at either det. 1 or det. 2 (i.e., qubit 1 low) indicates a constant function; each looks at an interferometer comparing the two halves of the oracle. Interfering 1 with 4 and 2 with 3 is as effective as interfering 1 with 3 and 2 with 4 -- but insensitive to this decoherence model. Schematic of DJ

30. Wait… Shift… Measure ground state population Quantum state reconstruction Initial phase- space distribution

31. Q(x,p) for a coherent H.O. state

32. Theory for 80/20 mix of e and g