Quantum information with photons and atoms from tomography

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 PQE 2003

Acknowledgments 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 Myrskog Ana Jofre Salvatore Maone Jalani Fox Mirco Siercke Samansa Maneshi TBA: Rob Adamson Theory friends: Daniel Lidar, Janos Bergou, John Sipe, Paul Brumer, Howard Wiseman

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…

Density matrices and superoperators

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

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.

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

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

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

Testing the superoperator LL = input state Predicted Nphotons = 297 ± 14

Testing the superoperator LL = input state Predicted Observed Nphotons = 297 ± 14 Nphotons = 314

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

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

Model beamsplitter predicitons Singlet filter Best Fit: 1 = 0. 76 π 2 = 0. 80 π Predicted

Comparison to measured Superop Observed Predicted

Performing a quantum computation in a decoherence-free subspace The Deutsch-Jozsa algorithm: A Oracle A H x H y y f(x) x H 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. Photon number basis 1 2 3 4 Computational basis 1 st qubit 2 nd qubit

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

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

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

Original Encoding DFS Encoding C B B C C Constant function B Balanced function

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

Houston, we have separation!

Quantum state reconstruction p p = x t x Initial phasespace distribution x Wait… Shift… p Q(0, 0) = Pg (More recently: direct density-matrix reconstruction) x Measure ground state population

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

Exp't: "W" or [Pg-Pe](x, p)

W(x, p) for 80% excitation

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 2 x 1 -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

Schematic diagram of D-J interferometer 1 2 3 4 Oracle 1 00 2 01 3 10 4 11 1 2 3 4 “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.

Quantum state reconstruction Wait… Measure ground state population Shift… Initial phasespace distribution

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

Theory for 80/20 mix of e and g