Abstract

Distributed quantum information processing requires a reliable quantum memory and a faithful carrier of quantum information. Atomic qubits have very long coherence times and are thus excellent candidates for quantum information storage, whereas photons are ideal for the transport of quantum information as they can travel long distances with a minimum of decoherence. We discuss the theoretical and experimental combination of these two systems and their use for not only quantum information transfer but also scalable quantum computation architectures.

© 2007 Optical Society of America

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2006 (9)

W. K. Hensinger, S. Olmschenk, D. Stick, D. Hucul, M. Yeo, M. Acton, L. Deslauriers, C. Monroe, and J. Rabchuk, "T-junction ion trap array for two-dimensional ion shuttling, storage and manipulation," Appl. Phys. Lett. 88, 034101 (2006).
[CrossRef]

D. Stick, W. K. Hensinger, S. Olmschenk, M. J. Madsen, K. Schwab, and C. Monroe, "Ion trap in a semiconductor chip," Nat. Phys. 2, 36-39 (2006).
[CrossRef]

L. Deslauriers, S. Olmschenk, D. Stick, W. K. Hensinger, J. Sterk, and C. Monroe, "Scaling and suppression of anomalous heating in ion traps," Phys. Rev. Lett. 97, 103007 (2006).
[CrossRef] [PubMed]

L.-M. Duan, M. J. Madsen, D. L. Moehring, P. Maunz, R. N. Kohn, Jr., and C. Monroe, "Probabilistic quantum gates between remote atoms through interference of optical frequency qubits," Phys. Rev. A 73, 062324 (2006).
[CrossRef]

M. J. Madsen, D. L. Moehring, P. Maunz, R. N. Kohn, Jr., L.-M. Duan, and C. Monroe, "Ultrafast coherent coupling of atomic hyperfine and photon frequency qubits," Phys. Rev. Lett. 97, 040505 (2006).
[CrossRef] [PubMed]

J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer, and H. Weinfurter, "Observation of entanglement of a single photon with a trapped atom," Phys. Rev. Lett. 96, 030404 (2006).
[CrossRef] [PubMed]

D. L. Moehring, B. B. Blinov, D. W. Gidley, R. N. Kohn, Jr., M. J. Madsen, T. B. Sanderson, R. S. Vallery, and C. Monroe, "Precision lifetime measurement of a single trapped ion with ultrafast laser pulses," Phys. Rev. A 73, 023413 (2006).
[CrossRef]

M. Acton, K.-A. Brickman, P. Haljan, P. J. Lee, L. Deslauriers, and C. Monroe, "Near-perfect simultaneous measurement of a qubit register," Quantum Inf. Comput. 6, 465-482 (2006).

J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquie, G. Messin, A. Browaeys, and P. Grangier, "Quantum interference between two single photons emitted by independently trapped atoms," Nature 440, 779-782 (2006).
[CrossRef] [PubMed]

2005 (9)

C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. DeMarco, A. Ben-Kish, R. B. Blakestad, J. Britton, D. B. Hume, W. M. Itano, D. Leibfried, R. Reichle, T. Rosenband, T. Schaetz, P. O. Schmidt, and D. J. Wineland, "Long-lived qubit memory using atomic ions," Phys. Rev. Lett. 95, 060502 (2005).
[CrossRef] [PubMed]

H. Häffner, F. Schmidt-Kaler, W. Hänsel, C. F. Roos, T. Körber, M. Chwalla, M. Riebe, J. Benhelm, U. D. Rapol, C. Becher, and R. Blatt, "Robust entanglement," Appl. Phys. B 81, 151-153 (2005).
[CrossRef]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, "Noise-free high-efficiency photon-number-resolving detectors," Phys. Rev. A 71, 061803 (2005).
[CrossRef]

L.-M. Duan and R. Raussendorf, "Efficient quantum computation with probabilistic quantum gates," Phys. Rev. Lett. 95, 080503 (2005).
[CrossRef] [PubMed]

S. D. Barrett and P. Kok, "Efficient high-fidelity quantum computation using matter qubits and linear optics," Phys. Rev. A 71, 060310(R) (2005).
[CrossRef]

C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, "Measurement-induced entanglement for excitation stored in remote atomic ensembles," Nature 438, 828-832 (2005).
[CrossRef] [PubMed]

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P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan, L.-M. Duan, and C. Monroe, "Phase control of trapped ion quantum gates," J. Opt. B: Quantum Semiclassical Opt. 7, S371-S383 (2005).
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2004 (9)

L.-M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, "Scaling trapped ions for quantum computation with probabilistic ion-photon mapping," Quantum Inf. Comput. 4, 165-173 (2004).
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J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, "Deterministic generation of single photons from one atom trapped in a cavity," Science 303, 1992-1994 (2004).
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B. B. Blinov, D. Leibfried, C. Monroe, and D. J. Wineland, "Quantum computing with trapped ion hyperfine qubits," Quantum Inf. Process. 3, 45-59 (2004).
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A. P. VanDevender and P. G. Kwiat, "High efficiency single photon detection via frequency up-conversion," J. Mod. Opt. 51, 1433-1445 (2004).
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B. B. Blinov, D. L. Moehring, L.-M. Duan, and C. Monroe, "Observation of entanglement between a single trapped atom and a single photon," Nature 428, 153-157 (2004).
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D. L. Moehring, M. J. Madsen, B. B. Blinov, and C. Monroe, "Experimental Bell inequality violation with an atom and a photon," Phys. Rev. Lett. 93, 090410 (2004).
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D. N. Matsukevich and A. Kuzmich, "Quantum state transfer between matter and light," Science 306, 663-666 (2004).
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2003 (14)

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
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J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nägerl, D. M. Stamper-Kurn, and H. J. Kimble, "State-insensitive cooling and trapping of single atoms in an optical cavity," Phys. Rev. Lett. 90, 133602 (2003).
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F. Troiani, E. Molinari, and U. Hohenester, "High-finesse optical quantum gates for electron spins in artificial molecules," Phys. Rev. Lett. 90, 206802 (2003).
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T. Calarco, A. Datta, P. Fedichev, E. Pazy, and P. Zoller, "Spin-based all-optical quantum computation with quantum dots: understanding and suppressing decoherence," Phys. Rev. A 68, 012310 (2003).
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L.-M. Duan and J. Kimble, "Efficient engineering of multiatom entanglement through single-photon detections," Phys. Rev. Lett. 90, 253601 (2003).
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J. Kim, C. J. Nuzman, B. Kumar, D. F. Lieuwen, J. S. Kraus, A. Weiss, C. P. Lichtenwalner, A. R. Papazian, R. E. Frahm, N. R. Basavanhally, D. A. Ramsey, V. A. Aksyuk, F. Pardo, M. E. Simon, V. Lifton, H. B. Chan, M. Haueis, A. Gasparyan, H. R. Shea, S. Arney, C. A. Bolle, P. R. Kolodner, R. Ryf, D. T. Neilson, and J. V. Gates, "1100×1100 port MEMS-based optical crossconnect with 4-dB maximum loss," IEEE Photon. Technol. Lett. 15, 1537-1539 (2003).
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M. Barrett, B. L. DeMarco, T. Schaetz, V. Meyer, D. Leibfried, J. Britton, J. Chiaverini, W. M. Itano, B. M. Jelenkovic, J. D. Jost, C. Langer, T. Rosenband, and D. J. Wineland, "Sympathetic cooling of 9Be+ and 24Mg+ for quantum logic," Phys. Rev. A 68, 042302 (2003).
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2002 (14)

B. B. Blinov, L. Deslauriers, P. Lee, M. J. Madsen, R. Miller, and C. Monroe, "Sympathetic cooling of trapped Cd+ isotopes," Phys. Rev. A 65, 040304(R) (2002).
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C. Monroe, "Quantum information processing with atoms and photons," Nature 416, 238-246 (2002).
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N. Lütkenhaus and M. Jahma, "Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack," New J. Phys. 4, 44 (2002).
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D. G. Enzer, P. G. Hadley, R. J. Hughes, C. G. Peterson, and P. G. Kwiat, "Entangled-photon six-state quantum cryptography," New J. Phys. 4, 45 (2002).
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C. Elliott, "Building the quantum network," New J. Phys. 4, 46 (2002).
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J. H. Shapiro, "Architectures for long-distance quantum teleportation," New J. Phys. 4, 47 (2002).
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D. Kielpinski, C. Monroe, and D. Wineland, "Architecture for a large-scale ion-trap quantum computer," Nature 417, 709-711 (2002).
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C. Piermarocchi, P. Chen, L. J. Sham, and D. G. Steel, "Optical RKKY interaction between charged semiconductor quantum dots," Phys. Rev. Lett. 89, 167402 (2002).
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A. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, "Coupling a single atomic quantum bit to a high finesse optical cavity," Phys. Rev. Lett. 89, 103001 (2002).
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A. Kuhn, M. Hennrich, and G. Rempe, "Deterministic single-photon source for distributed quantum networking," Phys. Rev. Lett. 89, 067901 (2002).
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2001 (4)

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
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L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, "Long-distance quantum communication with atomic ensembles and linear optics," Nature 414, 413-418 (2001).
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2000 (12)

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, "Fast quantum gates for neutral atoms," Phys. Rev. Lett. 85, 2208-2211 (2000).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
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G. J. Milburn, S. Schneider, and D. F. V. James, "Ion trap quantum computing with warm ions," Fortschr. Phys. 48, 801-810 (2000).
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D. Jonathan, M. B. Plenio, and P. L. Knight, "Fast quantum gates for cold trapped ions," Phys. Rev. A 62, 042307 (2000).
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J. L. Cirac and P. Zoller, "A scalable quantum computer with ions in an array of microtraps," Nature 404, 579-581 (2000).
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C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, "The atom-cavity microscope: single atoms bound in orbit by single photons," Science 287, 1447-1453 (2000).
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A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continuous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000).
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A. Imamoglu, "Quantum computation using quantum dot spins and microcavities," Fortschr. Phys. 48, 987-997 (2000).
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1999 (4)

C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez, and P. Zoller, "Creation of entangled states of distant atoms by interference," Phys. Rev. A 59, 1025-1033 (1999).
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L. Mandel, "Quantum effects in one-photon and two-photon interference," Rev. Mod. Phys. 71, S274-S282 (1999).
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1998 (1)

W. M. Itano, J. C. Bergquist, J. J. Bollinger, D. J. Wineland, U. Eichmann, and M. G. Raizen, "Complementarity and Young's interference fringes from two atoms," Phys. Rev. A 57, 4176-4187 (1998).
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1997 (1)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, "Quantum state transfer and entanglement distribution among distant nodes in a quantum network," Phys. Rev. Lett. 78, 3221-3224 (1997).
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1996 (1)

R. G. DeVoe and R. G. Brewer, "Observation of superradiant and subradiant spontaneous emission of two trapped ions," Phys. Rev. Lett. 76, 2049-2052 (1996).
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1995 (2)

J. I. Cirac and P. Zoller, "Quantum computation with cold trapped ions," Phys. Rev. Lett. 74, 4091-4094 (1995).
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C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, "Demonstration of a fundamental quantum logic gate," Phys. Rev. Lett. 75, 4714-4717 (1995).
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1993 (2)

P. G. Kwiat, A. M. Steinberg, R. Y. Chiao, P. Eberhard, and M. Petroff, "High efficiency single-photon detectors," Phys. Rev. A 48, R867-R870 (1993).
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1991 (1)

J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, "A 303 MHz frequency standard based on trapped Be+ ions," IEEE Trans. Instrum. Meas. 40, 126-128 (1991).
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1988 (2)

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1987 (2)

F. Diedrich and H. Walther, "Nonclassical radiation of a single stored ion," Phys. Rev. Lett. 58, 203-206 (1987).
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C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
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1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, "SU(2) and SU(1,1) interferometers," Phys. Rev. A 33, 4033-4054 (1986).
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1982 (1)

A. Aspect, P. Grangier, and G. Roger, "Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: a new violation of Bell's inequalities," Phys. Rev. Lett. 49, 91-94 (1982).
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1972 (1)

S. J. Freedman and J. F. Clauser, "Experimental test of local hidden-variable theories," Phys. Rev. Lett. 28, 938-941 (1972).
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W. K. Hensinger, S. Olmschenk, D. Stick, D. Hucul, M. Yeo, M. Acton, L. Deslauriers, C. Monroe, and J. Rabchuk, "T-junction ion trap array for two-dimensional ion shuttling, storage and manipulation," Appl. Phys. Lett. 88, 034101 (2006).
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M. Acton, K.-A. Brickman, P. Haljan, P. J. Lee, L. Deslauriers, and C. Monroe, "Near-perfect simultaneous measurement of a qubit register," Quantum Inf. Comput. 6, 465-482 (2006).

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J. Kim, C. J. Nuzman, B. Kumar, D. F. Lieuwen, J. S. Kraus, A. Weiss, C. P. Lichtenwalner, A. R. Papazian, R. E. Frahm, N. R. Basavanhally, D. A. Ramsey, V. A. Aksyuk, F. Pardo, M. E. Simon, V. Lifton, H. B. Chan, M. Haueis, A. Gasparyan, H. R. Shea, S. Arney, C. A. Bolle, P. R. Kolodner, R. Ryf, D. T. Neilson, and J. V. Gates, "1100×1100 port MEMS-based optical crossconnect with 4-dB maximum loss," IEEE Photon. Technol. Lett. 15, 1537-1539 (2003).
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It is also possible to excite on the D2 line to the 2P3/2 manifold (for I=1/2) where the ∣up arrow› and ∣down arrow› states are excited to ∣F′=1, mF′=0› and ∣F′=2, mF′=0›, respectively.

P. Maunz, M. J. Madsen, D. L. Moehring, R. N. Kohn, Jr., K. Younge, and C. Monroe, "Quantum interference of photon pairs from two trapped atomic ions," arXiv.org e-Print archive, August 5, 2006, quant-ph/0608047, http://arXiv.org/labs/quant-ph/0608047.

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Figures (13)

Fig. 1
Fig. 1

Possible schemes for atom–photon entanglement. (a) Energy level diagram for an atom with nuclear spin I = 1 2 and magnetic moment μ I < 0 . (b) Decay scheme unique to the P 3 2 2 level with two possible decay channels. If the photon is emitted perpendicularly to the quantization axis, the polarization modes are linear and orthogonal. (c) Decay scheme consisting of three decay channels where viewing along the quantization axis eliminates the photon from the Δ m = 0 decay channel due to the radiation pattern, and the Δ m = ± 1 photons have orthogonal circular polarizations. (d) Same decay scheme as (c) but viewed perpendicularly to the quantization axis. The Δ m = 0 photon decay channel is linear and orthogonal to the Δ m = ± 1 decay channels. After decay, the 1 , 1 and ∣1,1⟩ can be coherently combined in the ∣0,0⟩ state establishing the atomic qubit. (e) Two Δ m = 1 decay channels with the same polarization comprise a photonic frequency qubit. The Δ m = 0 photon can be eliminated by a polarizer or by the radiation pattern if viewed along the quantization axis. (f) Two Δ m = 0 decay channels with the same polarization and different frequencies. Viewed perpendicularly to the quantization axis, the Δ m = ± 1 photons are eliminated via a polarizer. As described in the text, this decay scheme can be used to perform quantum gates between the atom and the photon. [Note that cases (c)–(f) also apply to the P 3 2 2 levels.] (g) Radiation emission patterns for the Δ m = 0 and Δ m = ± 1 decay channels defined by a magnetic field B .

Fig. 2
Fig. 2

Entanglement schemes for two remotely located atomic qubits. (a) In the single-photon protocol by Cabrillo et al. (Ref. [55]), each atom is weakly excited with a probability ϵ from the ground state ∣↓⟩ to the excited state e . The photon detectors are only sensitive to the e to ∣↑⟩ decay, and a detection of a single photon projects the atoms into the entangled state ( a b + e i ϕ a b ) 2 . (b) In the two-photon protocol, each atom is prepared in the excited state with two decay channels giving rise to two distinguishable photonic qubit states. (c) Detection setup suitable for either entanglement protocol. With the protocol by Cabrillo et al., only one of the two detectors detects a photon; whereas the two-photon protocol requires coincident detection on each detector projecting the atoms into the entangled Bell state Ψ atom = ( a b a b ) 2 . (d) Alternative setup when using polarization qubits in the two-photon protocol. Coincident detection between D1 and D3 or D2 and D4 projects the atoms into the state Ψ atom , whereas coincident detection between D1 and D2 or D3 and D4 results in Ψ + atom = ( a b + a b ) 2 .

Fig. 3
Fig. 3

Energy levels (not to scale) and laser bandwidth requirements for the generation of a high efficiency single-photon source (atomic values shown are for Cd + 111 ). For simultaneous excitation of all hyperfine states, the bandwidth of the laser pulse must be much larger than the largest hyperfine splitting but smaller than the fine structure splitting to eliminate coupling to the different excited state levels. To eliminate multiple scatters, the pulse duration must be much smaller than the excited state lifetime (have a bandwidth much larger than the linewidth of the excited state).

Fig. 4
Fig. 4

(a) Spatial modes a and b are straight paths through the BS, and the BS interferes with these two modes. (b)–(e) Four possible output modes of two photons entering a BS from different ports. A negative phase is acquired only upon reflection from low to high index of refraction—mode a in (c) and (e).

Fig. 5
Fig. 5

Experimental procedure for atom–photon entanglement with photon polarization qubits (Refs. [73, 74]). (a) The atom is initialized to the ∣1,0⟩ state via optical pumping to the ∣0,0⟩ state and a microwave transfer pulse. (b) The atom is driven to the P 3 2 2 ∣2,1⟩ excited state resulting in spontaneous emission via two photon decay channels of orthogonal polarizations when viewed perpendicularly to the quantization axis. The resulting entangled state is 1 3 H + 2 3 V . (c) Microwave ( μ wave ) pulse resonant with the 1 , 1 0 , 0 transition coherently transfers the population to the clock qubit states. (d) Second microwave pulse prepares the atomic qubit for measurement in any basis. (e) σ + -polarized laser pulse performs the state detection of the atomic qubit using resonance fluorescence techniques.

Fig. 6
Fig. 6

Experimental procedure for atom–photon entanglement with photon frequency qubits (Ref. [49]). (a) The ion is initialized in the state ( + ) 2 via optical pumping to the ∣0,0⟩ state and a microwave π 2 pulse. (b) Superposition of atomic qubit states is coherently driven to the P 3 2 2 excited state via a resonantly tuned π-polarized ultrafast laser pulse. (c) Second pulse drives the qubit back to the ground state a short time later. (d) Second π 2 microwave pulse with variable phase completes the Ramsey experiment, and the atomic state is measured using a resonance fluorescence technique. (e) Results from the microwave Ramsey experiment. Circles show the near perfect Ramsey fringes for the case with no ultrafast laser pulse. With a single ultrafast laser pulse, the coherence is lost due to the spontaneous emission of a photon that is not measured in a controlled, precisely timed fashion (squares). The average population in the bright state is above 0.5 due to the fluorescence branching ratios [Fig. 10 (inset)]. Upon application of a second ultrafast laser pulse, the coherence in the ion is maintained by driving the qubit states back down to the ground states (triangles).

Fig. 7
Fig. 7

Detection system for the two-photon interference experiment. The light from the two ions is separated on a BS and mode matched on the second BS. The photons are detected on single-photon sensitive PMTs. A camera is used for coarse alignment, and the nonoverlapping photon modes are blocked by irises.

Fig. 8
Fig. 8

Intensity autocorrelation for cw excitation. The dashed curve shows strong antibunching for a single ion with g 1 ( 2 ) ( 0 ) = 0.18 limited by the resolution of the detection system. With this value, the expected antibunching of light from two nonoverlapping ions is expected to be g 2 , u m ( 2 ) ( 0 ) = 0.59 in good agreement with the experimental value (dotted curve). If the two-photon modes are matched, the interference leads to a significant reduction of coincidence detections (solid curve). The measured antibunching was g 2 , m ( 2 ) ( 0 ) = 0.31 corresponding to a mode overlap of 57 % .

Fig. 9
Fig. 9

Intensity autocorrelation of the light emitted by a single ion excited by an ultrafast laser. The near-perfect antibunching at t = 0 shows that at most one photon is emitted from an excitation pulse.

Fig. 10
Fig. 10

Ion bright state population as a function of pulse energy. Each point represents a collection of 60,000 runs. As the population in the excited P state is driven to unity, the bright state population approaches 1 3 (horizontal dashed line) determined by the spontaneous emission branching ratio. The data are fit to a single parameter giving a value a = 0.42 pJ 1 2 . Inset: Relevant energy levels for the S P Rabi oscillation experiment. A π-polarized ultrafast laser pulse excited the ion from the ground state to the excited state with variable energy. The three possible decay channels are shown with their respective fluorescence branching ratios. After a time ( 10 μ s ) following the excitation pulse, the bright state population of the ion was measured using resonance fluorescence detection.

Fig. 11
Fig. 11

Entanglement device capable of entangling multiple atoms simultaneously using micromirror arrays (Ref. [89]). Any pair of atoms can be entangled by routing the emitted photon from each atom to a BS, where single-photon detections can project the atoms into an entangled state. Parallelism is possible with this setup for N atoms with 2 N mirrors and N 2 BS pairs.

Fig. 12
Fig. 12

(a) Schematic of the quantum computation model based on probabilistic photon-mediated entanglement between atoms. The ancilla atoms ( i , j ) are entangled through the probabilistic protocols described in the text, and deterministic gates on the logic atoms ( i , j ) are constructed from local motional gates and probabilistic remote ancilla entanglement. (b) Schematic of quantum repeaters with trapped atoms based on probabilistic remote entanglement and local Coulomb interactions.

Fig. 13
Fig. 13

Illustration of the necessary steps for the construction of cluster states. (a) Controlled phase flip (CPF) entangling gate is used to extend the length of a 1D cluster. (b) Construction of a cross-shaped cluster from two 1D cluster chains. A Hadamard gate (H) is applied on the middle qubit of one chain, and a CPF gate connects the two middle qubits. Finally, an X measurement on one middle qubit removes the extra atom. (c), (d) Construction of a square lattice cluster state from the cross-shaped cluster states. CPF gates combine the shapes along ends of the crosses and X measurements are used to remove the remaining redundant qubits.

Tables (2)

Tables Icon

Table 1 Mixed Quantum State of a Single Atom and a Single Photon of Spatial Mode s after Time T rep Longer than the Spontaneous Emission Lifetime of the Atom a

Tables Icon

Table 2 Mixed Quantum State of Two Photon Modes and Two Atomic Qubits Given that Two Atom–Photon Systems are Prepared According to Table 1 and the Photon Modes are Coupled on a BS a

Equations (16)

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Ψ = i , j , Δ m C i , j , Δ m S i ν j Π Δ m ,
1 2 ( a γ a + a γ a ) ( b γ b + b γ b ) = 1 2 ( Ψ + atom Ψ + photon + Ψ atom Ψ photon + Φ + atom Φ + photon + Φ atom Φ photon ) ,
n a m b e i χ J ̂ y n a m b ,
( 0 a 1 b 1 a 0 b ) [ cos χ 2 sin χ 2 sin χ 2 cos χ 2 ] ( 0 a 1 b 1 a 0 b ) .
( 0 a 2 b 1 a 1 b 2 a 0 b ) [ 1 2 ( 1 + cos χ ) 1 2 sin χ 1 2 ( 1 cos χ ) 1 2 sin χ cos χ 1 2 sin χ 1 2 ( 1 cos χ ) 1 2 sin χ 1 2 ( 1 + cos χ ) ] ( 0 a 2 b 1 a 1 b 2 a 0 b ) .
Ψ ent = cos θ s s 0 s α 1 s β + sin θ s s 1 s α 0 s β ,
Ψ β α a b = N 1 ( cos θ a sin θ b cos 2 χ 2 a b sin θ a cos θ b sin 2 χ 2 a b ) ,
Ψ α β a b = N 2 ( cos θ a sin θ b sin 2 χ 2 a b sin θ a cos θ b cos 2 χ 2 a b ) ,
Ψ α β a a = N 3 [ ( cos θ a sin θ b a b + sin θ a cos θ b a b ) sin χ ] ,
Ψ α β b b = Ψ α β a a ,
Ψ ent diff = N ( cos θ a sin θ b a b sin θ a cos θ b a b ) ,
Ψ ent same = N ( cos θ a sin θ b a b + sin θ a cos θ b a b ) ,
ρ post = P good Ψ ent Ψ ent + P bad M a M b M a M b ,
P good = 1 4 p 2 p e 2 ( cos 2 θ a sin 2 θ b + sin 2 θ a cos 2 θ b ) ,
P bad = p 2 2 [ p b g ( 3 4 p b g + p e ) + p 2 e ( p b g + p e + 1 2 ) ] .
F = P good + 1 4 P bad P good + P bad = p e 2 + [ p b g ( 3 4 p b g + p e ) + p 2 e ( p b g + p e + 1 2 ) ] p e 2 + 4 [ p b g ( 3 4 p b g + p e ) + p 2 e ( p b g + p e + 1 2 ) ] .

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