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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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