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					                                            Efficient Unitary Quantum
                                            Memory in Atomic Vapour
                                                      Systems

                                                                                       Karl Surmacz

                                                                               (University of Oxford, UK)




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                       Overview


   •        Motivation – what, where, and why?


   •        Possible implementations


   •        Entanglement fidelity of a quantum memory


   •        Optimal read-in for a Raman quantum memory


   •        Optimal read-out


   •        Outlook


Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                       Motivation

   •        One use for a quantum memory (QM) – quantum repeater.

   •        Alice and Bob want to share entangled qubits over a large distance:




   •        Solution1 – generate entanglement over short distances, and propagate:

                                                                                                    • Keep fidelity above a certain
                                                                                                    threshold to give entangled pair
                                                                                                    over long distance with arbitrarily
                                                                                                    high fidelity.
                                                                                   Purification
                                                                                                    • Swap (teleportation) requires
                                                                                                    classical communication. Need to
                                                                                   “Swapping”
                                                                                                    store qubits in meantime.


                                                                                                    1. Briegel et al. PRL 81, 5932 (1998).
Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                  How it works

   •        Store quantum signal field                                             as an
            excitation  in a medium.




   •        Three-level system ensures “on
            demand” retrieval, no spontaneous
            emission.

   •        Storage is affected by some strong
            classical control field.

   •        |1> and |3> should be non-degenerate
            for addressability.

   •        Two approaches
             – Absorption
             – State preparation2 (NBI).

       2. C. Muschik et al., Phys. Rev. A 73 062329 (2006).


Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                          Implementations


       •        Single atom schemes
                 – Λ atom in a cavity3.
                 – Difficult to achieve required coupling,
                    susceptible to atom loss (!).

       •        Atomic ensembles – coupling enhanced by                                     .

                   – Storage state ~

                   – Specific schemes
                               • On resonant EIT schemes4,5 (Harvard,
                                 Georgia).
                               • Off-resonant Raman (Oxford).                          3. Boozer et al., quant-ph/0702248 (2007).

                               • CRIB (Geneva)6                                        4. Fleischhauer & Lukin, PRA 65, 022314 (2000),
                                                                                         Gorshkov et al., quant-ph/0604037 (2006).
                                                                                       5. Chaneliere et al., Nature 438, 833 (2005).
                                                                                       6. Kraus et al., PRA 73, 020302(R) (2006).


Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                          Fidelity of a General QM
                                                                       K. Surmacz et al., PRA 74, 050302(R) (2006).



   •        Assume a QM to consist of N two-level
            absorbers – appiles to all absorptive
            schemes.

   •        Qubit (q = 0,1 represent logical states)
            encoded in a photon, annihilation operator
            .

   •        Quantum repeater requires preservation of
            entanglement - entanglement fidelity of
            channel     .



                                                                                               •   Photonic qubit entangled with auxiliary
                                                                                                   qubit.

                                                                                               •   Atoms initially uncorrelated.




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                               Entanglement Fidelity

   •        Entanglement fidelity of a quantum
            memory:




                     •        Unitary allows for evolution of photon that does not decrease
                              entanglement – the „ideal memory‟ that maximizes fidelity.



                     •        ΛM consists of read-in, storage time ts, and read-out.



Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                             Calculating the fidelity

   •        N absorbers initially in state



   •        Hamiltonians for interaction for single atom:




  •        Couplings




  •        In practice pulses – assume properly matched
           (later), consider simple time dependence.




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                      Absorber-dependent parameters

   •        K and M broadenings during interaction
            time
              – Raman: due to Doppler broadening.

   •        κ magnitude of coupling – may change
            from atom-to-atom.
              – Raman: Doppler changes detunings.

   •        f(ts) storage time dephasing.
              – Raman: Atomic motion

   •        δa and δb model fact that each atom may
            couple to slightly different mode.
             – Raman: Doppler causes atoms to see
                 different signal frequency.

   •        Treat parameters as normally-distributed
            stochastic variables, e.g. with mean    ,
            width     .



Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                      Calculating fidelity

   •        Treat fluctuations in K and M
            perturbatively, assume                                                     .

   •        Solve system for final wavefunction,
            then construct full state of memory
            qubit.

   •        Average stochastic variables over
            ensemble.

   •        Minimizing over all                                             gives
            entanglement fidelity:




        •        Broadening terms scale as 1/N.



Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                              Measuring this fidelity

   •        Storing both logical states in one
            ensemble is difficult (see later for one
            possibility).

   •        Assume both logical states stored and
            retrieved in same way, then can
            measure fidelity experimentally.


                                                                                       D2   •   Source of separable photons.

                                                                                            •   Tune the pulse shaper so that
                                                                                                coincidence detections are minimum.
 source
                                                  PS                                        •   This corresponds to when the PS
                                                                                                gives the mode of memory output with
                                                                                       D1       largest eigenvalue (i.e.   )



Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                       Raman QM
                                                                            J. Nunn et al., PRA 74, 011401(R) (2007).



   •        In general, signal pulse will have
            temporal shape – shape control field for
            optimal storage.

   •        Long thin ensemble - 1D propagation.
            Signal group velocity vs, signal and
            control wavevectors (frequencies) ks and
            kc (ωs and ωc).


                                                                                                •    Signal pulse given by operator




                                                                                                •    „Spin wave‟ (atomic excitation)




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                       Raman QM

   •        Polarization P of ensemble given by
            dipole moment operator:




   •        P obtained by summing up contributions
            for all atoms in slice


                                                                                            •   Maxwell-Bloch equations come from
                                                                                                 – Atomic evolution




                                                                                                 – Maxwell equation




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                       Maxwell-Bloch equations

  •         >> photon bandwidth - adiabatically
           eliminate upper state.

  •        Slowly-varying and paraxial approximations.

  •        Overall coupling C, control pulse area E.
           Maxwell-Bloch equations for slowly-varying
           variables:




                                                                                       •   Eliminate control field pulse
                                                                                           envelope       :




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                            Optimal storage



•       Solution for read-in (~ -> ):




•       Use SVD to match control - signal input
        is right singular vector associated with
        largest singular value of M.


•       Storage can be achieved with 99%
        efficiency for a finite coupling C=2.

•       n~1019m-3, ~1013Hz, c~ 1010Hz.




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                       Retrieval Problem

   •        Resulting B highly asymmetric.

   •        Retrieval process time-reverse of
            storage. B for optimal forward
            readout = mirror image of optimally
            stored B.




                                                                                       •   Solution - backwards read-out. But:




                                                                                       •   Phase mismatch -> degrades
                                                                                           efficiency when integrating if |1>
                                                                                           and |3> non-degenerate.

Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                         Optimal Retrieval
                                                                             K. Surmacz et al. in preparation (2007).




   •        Solution – swap lower levels, angle
            control field to match ω13, so that
            longitudinal phases match.

   •        Enables optimal retrieval of stored signal
            field so that fields spatially separated.




                                                                                                        •    Limitations: angle must be small
                                                                                                             enough for field group velocities to be ~
                                                                                                             equal.


                  Unmatched                                                       Matched               •    Stokes shift limits signal field
                                                                                                             bandwidth.



Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                               Fidelity of Raman QM

   •        Recall:




   •        For Raman QM:




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                               Multimode QM

   •        Use phasematching to store a
            frequency-encoded qubit.

   •        Signal consisting of two photons,
            each with different frequency. Each
            photon requires different control field
            angle.

   •        Spin wave has two components, and
            photons emerge in different
            directions.

   •        For reasonable coupling, solution
            almost identical two single-photon
            processes.                                                                  •   Entanglement fidelity of
                                                                                            memory for qubit:



Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                       Summary



   •        General entanglement fidelity of quantum memory.

   •        Raman QM allows broadband pulse storage.

   •        Modematched Raman QM for optimal read-in.

   •        Phase mismatch prevents optimal pulse retrieval for non-degenerate atom

   •        Solve by angling control fields - enables storage of a frequency-encoded qubit.




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                                       Outlook


   •        Phasematching also satisfied for two
            control fields. Read-out gives splitting
            of output signal - beamsplitter?




   •        Quantum memory in optical lattice
             • No motion of absorbers
             • Different sources of error
             • Regular discrete geometry




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.
                                                                      Acknowledgments


                                                                     Oxford Quantum Memory Group:

                                                                                         Ian Walmsley
                                                                                         Dieter Jaksch
                                                                                       Zhongyang Wang
                                                                                         Joshua Nunn
                                                                                       Felix Waldermann
                                                                                             KC Lee

                                                                                           Money:




Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.

				
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