1. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Karl Surmacz (University of Oxford, UK) Efficient Unitary Quantum Memory in Atomic Vapour Systems

2. 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

3. 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: Solution 1 – generate entanglement over short distances, and propagate: Purification “Swapping” Keep fidelity above a certain threshold to give entangled pair over long distance with arbitrarily high fidelity. Swap (teleportation) requires classical communication. Need to store qubits in meantime. 1. Briegel et al. PRL 81, 5932 (1998).

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

5. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Implementations Single atom schemes –Λ atom in a cavity 3. –Difficult to achieve required coupling, susceptible to atom loss (!). Atomic ensembles – coupling enhanced by. –Storage state ~ –Specific schemes On resonant EIT schemes 4,5 (Harvard, Georgia). Off-resonant Raman (Oxford). CRIB (Geneva) 6 3. Boozer et al., quant-ph/0702248 (2007). 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).

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

7. 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. Λ M consists of read-in, storage time t s, and read-out.

8. 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.

9. 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(t s ) storage time dephasing. –Raman: Atomic motion δ a and δ b 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.

10. 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.

11. 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. D2D2 D1D1 source PS Source of separable photons. Tune the pulse shaper so that coincidence detections are minimum. This corresponds to when the PS gives the mode of memory output with largest eigenvalue (i.e. )

12. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Raman QM In general, signal pulse will have temporal shape – shape control field for optimal storage. Long thin ensemble - 1D propagation. Signal group velocity v s, signal and control wavevectors (frequencies) k s and k c (ω s and ω c ). Signal pulse given by operator ‘Spin wave’ (atomic excitation) J. Nunn et al., PRA 74, 011401(R) (2007).

13. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Maxwell-Bloch equations come from –Atomic evolution –Maxwell equation Raman QM Polarization P of ensemble given by dipole moment operator: P obtained by summing up contributions for all atoms in slice

14. 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 :

15. 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~10 19 m -3,  ~10 13 Hz,  c ~ 10 10 Hz.

16. 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.

17. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Optimal Retrieval Solution – swap lower levels, angle control field to match ω 13, 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. Stokes shift limits signal field bandwidth. UnmatchedMatched K. Surmacz et al. in preparation (2007).

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

19. 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:

20. 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.

21. 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

22. 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: