Abstract

The ability to filter unwanted light signals is critical to the operation of quantum memories based on neutral atom ensembles. Here we demonstrate an efficient frequency filter which uses a vapor cell filled with 85Rb and a buffer gas to attenuate both residual laser light and noise photons by nearly two orders of magnitude with little loss to the single photons associated with our cold 87Rb quantum memory. This simple, passive filter provides an additional 18 dB attenuation of our pump laser and erroneous spontaneous emissions for every 1 dB loss of the single photon signal. We show that the addition of a frequency filter increases the non-classical correlations and the retrieval efficiency of our quantum memory by ≈ 35%.

© 2015 Optical Society of America

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References

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2014 (1)

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

2013 (5)

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498(7455), 466–469 (2013).
[Crossref] [PubMed]

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87(3), 031801 (2013).
[Crossref]

X. Fernandez-Gonzalvo, G. Corrielli, B. Albrecht, M. Grimau, M. Cristiani, and H. de Riedmatten, “Quantum frequency conversion of quantum memory compatible photons to telecommunication wavelengths,” Opt. Express 21(17), 19473–19487 (2013).
[Crossref] [PubMed]

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

2012 (1)

2011 (1)

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

2010 (3)

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

B. Zhao, M. Muller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

2008 (1)

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

2007 (2)

S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, “Collisional decoherence during writing and reading quantum states,” Phys. Rev. A 75(4), 040101 (2007).
[Crossref]

J. Camparo, “The rubidium atomic clock and basic research,” Phys. Today 60(11), 33–39 (2007).
[Crossref]

2004 (5)

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

D. N. Matsukevich and A. Kuzmich, “Quantum state transfer between matter and light,” Science 306(5696), 663–666 (2004).
[Crossref] [PubMed]

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

2003 (1)

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(6941), 731–734 (2003).
[Crossref] [PubMed]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

2001 (1)

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

1998 (1)

H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Abstreiter, G.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Agarwal, A.

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Albrecht, B.

Altin, P. A.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

André, A.

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

Balic, V.

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Bashkansky, M.

Bennet, C. H.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution, and coin-tossing,” Proc. 1984 IEEE International Conference on Computers, Systems, and Signal Processing, 175–179 (1984).

Bernu, J.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Bichler, M.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Bishof, M.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Boca, A.

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(6941), 731–734 (2003).
[Crossref] [PubMed]

Boozer, A. D.

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(6941), 731–734 (2003).
[Crossref] [PubMed]

Bowen, W. P.

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(6941), 731–734 (2003).
[Crossref] [PubMed]

Braje, D.

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Brassard, G.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution, and coin-tossing,” Proc. 1984 IEEE International Conference on Computers, Systems, and Signal Processing, 175–179 (1984).

Briegel, H.-J.

H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Browaeys, A.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Buchler, B. C.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Camparo, J.

J. Camparo, “The rubidium atomic clock and basic research,” Phys. Today 60(11), 33–39 (2007).
[Crossref]

Cardoso, G. C.

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Champion, T.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Childress, L.

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

Chou, C. W.

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(6941), 731–734 (2003).
[Crossref] [PubMed]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press2013).

Cirac, J. I.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Corrielli, G.

Cristiani, M.

de Riedmatten, H.

Duan, L.-M.

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(6941), 731–734 (2003).
[Crossref] [PubMed]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

Ducommun, Y.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Dudin, Y. O.

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498(7455), 466–469 (2013).
[Crossref] [PubMed]

Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87(3), 031801 (2013).
[Crossref]

Dur, W.

H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Laser Physics (John Wiley & Sons, 2010).
[Crossref]

Eisaman, M. D.

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

England, D.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Evellin, C.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Fatemi, F. K.

Fernandez-Gonzalvo, X.

Fernholz, T.

S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, “Collisional decoherence during writing and reading quantum states,” Phys. Rev. A 75(4), 040101 (2007).
[Crossref]

Figueroa, E.

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

Finley, J. J.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Gatan, A.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Geng, J.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

Glorieux, Q.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Goda, S.

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Gopal, V.

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Grangier, P.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Grimau, M.

Hahn, C.

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

Hammerer, K.

B. Zhao, M. Muller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Harris, S.

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Hayes, D.

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

Heifetz, A.

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Heiss, D.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Hosseini, M.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Jaksch, D.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Jiang, L.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Jin, X.-M.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Jordaan, B.

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

Kessler, E. M.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

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(6941), 731–734 (2003).
[Crossref] [PubMed]

Komar, P.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Kroutvar, M.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Kumar, P.

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Kupchak, C.

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

Kuzmich, A.

Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87(3), 031801 (2013).
[Crossref]

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498(7455), 466–469 (2013).
[Crossref] [PubMed]

D. N. Matsukevich and A. Kuzmich, “Quantum state transfer between matter and light,” Science 306(5696), 663–666 (2004).
[Crossref] [PubMed]

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(6941), 731–734 (2003).
[Crossref] [PubMed]

Lam, P. K.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Langford, N. K.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Lee, K. C.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Li, L.

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498(7455), 466–469 (2013).
[Crossref] [PubMed]

Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87(3), 031801 (2013).
[Crossref]

Lukin, M. D.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

Manz, S.

S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, “Collisional decoherence during writing and reading quantum states,” Phys. Rev. A 75(4), 040101 (2007).
[Crossref]

Massou, F.

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

Matsukevich, D. N.

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

D. N. Matsukevich and A. Kuzmich, “Quantum state transfer between matter and light,” Science 306(5696), 663–666 (2004).
[Crossref] [PubMed]

Maunz, P.

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

Michelberger, P.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Laser Physics (John Wiley & Sons, 2010).
[Crossref]

Miroshnychenko, Y.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Mittiga, T.

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

Moehring, D. L.

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

Monroe, C.

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

Muller, M.

B. Zhao, M. Muller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Namazi, M.

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

Neuzner, A.

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press2013).

Nolleke, C.

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

Nunn, J.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Olmschenk, S.

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

Pan, J.-W.

S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, “Collisional decoherence during writing and reading quantum states,” Phys. Rev. A 75(4), 040101 (2007).
[Crossref]

Reim, K. F.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Reiserer, A.

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

Rempe, G.

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

Ritter, S.

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

Robins, N. P.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Schmiedmayer, J.

S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, “Collisional decoherence during writing and reading quantum states,” Phys. Rev. A 75(4), 040101 (2007).
[Crossref]

Schuh, D.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

Shahriar, M.S.

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Sparkes, B. M.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Sprague, M. R.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Srensen, A. S.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Steck, D.A

D.A Steck, Rubidium 87 d line data, 2001.

Sussman, B. J.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

Vurgaftman, I.

Walmsley, I. A.

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Wilk, T.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Wolters, J.

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

Ye, J.

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Yin, G.

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

Zhao, B.

B. Zhao, M. Muller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Zibrov, A. S.

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

Zoller, P.

B. Zhao, M. Muller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Int. J. Quantum Info. (1)

S. Olmschenk, D. Hayes, D. N. Matsukevich, P. Maunz, D. L. Moehring, and C. Monroe, “Quantum logic between distant trapped ions,” Int. J. Quantum Info. 08(01n02), 337–394 (2010).
[Crossref]

Nat. Phys. (1)

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Srensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10(8), 582–587 (2014).
[Crossref]

Nature (5)

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498(7455), 466–469 (2013).
[Crossref] [PubMed]

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature 432(7013), 81–84 (2004).
[Crossref] [PubMed]

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(6941), 731–734 (2003).
[Crossref] [PubMed]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref] [PubMed]

New J. Phys. (1)

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15(8), 085027 (2013).
[Crossref]

Opt. Commun. (1)

A. Heifetz, A. Agarwal, G. C. Cardoso, V. Gopal, P. Kumar, and M.S. Shahriar, “Super efficient absorption filter for quantum memory using atomic ensembles in a vapor,” Opt. Commun. 232(1–6), 289–293 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, “Collisional decoherence during writing and reading quantum states,” Phys. Rev. A 75(4), 040101 (2007).
[Crossref]

Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87(3), 031801 (2013).
[Crossref]

B. Zhao, M. Muller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic rydberg gates,” Phys. Rev. A 81(5), 052329 (2010).
[Crossref]

Phys. Rev. Lett. (5)

C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe, and S. Ritter, “Efficient teleportation between remote single-atom quantum memories,” Phys. Rev. Lett. 110(14), 140403 (2013).
[Crossref] [PubMed]

D. Braje, V. Balic, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93(18), 183601 (2004).
[Crossref] [PubMed]

H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

T. Wilk, A. Gatan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010).
[Crossref] [PubMed]

M. D. Eisaman, L. Childress, A. André, F. Massou, A. S. Zibrov, and M. D. Lukin, “Shaping quantum pulses of light via coherent atomic memory,” Phys. Rev. Lett. 93(23), 233602 (2004).
[Crossref] [PubMed]

Phys. Today (1)

J. Camparo, “The rubidium atomic clock and basic research,” Phys. Today 60(11), 33–39 (2007).
[Crossref]

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

Science (2)

D. N. Matsukevich and A. Kuzmich, “Quantum state transfer between matter and light,” Science 306(5696), 663–666 (2004).
[Crossref] [PubMed]

K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, “Entangling macroscopic diamonds at room temperature,” Science 334(6060), 1253–1256 (2011).
[Crossref] [PubMed]

Other (5)

C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. Nolleke, and E. Figueroa, “Room-temperature quantum memory for polarization states,” arXiv:1405.6117 [quant-ph] (2014).

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution, and coin-tossing,” Proc. 1984 IEEE International Conference on Computers, Systems, and Signal Processing, 175–179 (1984).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press2013).

P. W. Milonni and J. H. Eberly, Laser Physics (John Wiley & Sons, 2010).
[Crossref]

D.A Steck, Rubidium 87 d line data, 2001.

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Figures (6)

Fig. 1
Fig. 1 The relevant energy levels and transitions of 87Rb and 85Rb used in this paper. The write beam is 20, 40, or 60 MHz blue-detuned from the 5S1/2,F = 2→5P1/2,F = 2 (D1) transition. Atoms excited to the 5P1/2,F = 2 state can decay to either the 5S1/2,F = 1 or the 5S1/2,F = 2 ground state in 87Rb. The write beam and the emission from the 5P1/2,F = 2 5S1/2,F = 2 state are detuned from the Doppler-broadened 5S1/2,F = 3 5P1/2 transition in 85Rb by ≈ 790 MHz. The emission from the 5P1/2,F = 2 5S1/2,F = 1 state is detuned from the Doppler-broadened 5S1/2,F = 2 5P1/2 transition in 85Rb by ≈ 3000 MHz. This difference in the detuning of these transitions serves as the basis for using a 85Rb vapor cell as a frequency filter in our quantum memory experiment. The read beam is resonant with the 5S1/2,F = 1 5P3/2,F = 2 (D2) transition in 87Rb. Drawing not to scale.
Fig. 2
Fig. 2 Plots of the transmitted light through the vapor cell filter as a function of temperature in the Doppler-broadened limit (Eq. (2)) for a filter cell length of 1 inch. (a) The pure 85Rb filter cell acts nearly perfectly by attenuating the undesired light strongly (blue, dotted) and no attenuation of the signal (magenta, dashed) leading to large difference in attenuation between the two signals (gold, solid). (b) Small concentrations of 87Rb (87Rb/85Rb fraction of 0.2%) result in strong attenuation of the resonant background and signal fields. The emission from the cold 87Rb ensemble is assumed to be hyperfine specific but the absorption by the warm 85Rb vapor cell transitions are assumed to be Doppler-broadened and not hyperfine specific (See Fig. 1).
Fig. 3
Fig. 3 Normalized measured transmission of classical laser light at the Noise (circles) and Signal (squares) frequencies and the difference between the two signals (diamonds) at a variety of vapor cell temperatures with 47 Torr of Ar in a 1 inch long 85Rb vapor cell. The corresponding solid lines are calculations of the transmitted light through the vapor cell filter as a function of temperature with 47 Torr of Ar using the filter response from Eq. (3).
Fig. 4
Fig. 4 (a) Counter-propagating Write and Read light pulses interact with a cold 87Rb ensemble while emitted single photons are detected off-axis by Si APDs. Arrival times of the photons are measured by a time-interval analyzer (TIA). Emitted Write photons are filtered by a warm 85Rb vapor cell to attenuate Noise photons. (b) A MOT is loaded and compressed to create a cold, dense cloud at a rate of 27 Hz. The MOT magnetic and laser fields are then extinguished for a 1 ms duration and a series of 600 Read-Write pulses interact with the system. (c) Log-Log plot of g A , B 2 as a function of pA with filter cell at 336 K and Δw/2π = 20 MHz. Write probability (pA) is varied from 10−2 − 10−4 and the corresponding values of g A , B 2 increase from ≈ 8 to > 160.
Fig. 5
Fig. 5 Probability of detecting a write photon and Retrieval Efficiency as a function of the filter cell temperature for Δw/2π =20, 40, and 60 MHz. (a) The write probability decreases by ≈ 40% over the range of filter cell temperatures used in this experiment for each write beam detuning. (b) A significant increase in Retrieval Efficiency as the filter cell temperature is increased is observed. The decrease in Noise photon detection in the Write arm leads to an ≈ 35% increase in the Retrieval Efficiency for Δw/2π = 40 and 60 MHz and 15% for Δw/2π = 20 MHz. Each point in plots (a) and (b) correspond to ~ 10000 write detection events and ~ 500 coincident counts respectively. The running time required to obtain a single data point is ≈ 12 minutes. Error bars are calculated assuming Poissonian statistics.
Fig. 6
Fig. 6 g A , B ( 2 ) correlations as a function of filter cell temperature for Δw/2π =20, 40 and 60 MHz. The strength of the correlations increase by ≈ 15% for Δw/2π = 20 MHz, ≈ 35% for Δw/2π = 40 MHz, and ≈ 35% for Δ/2π = 40 MHz. The increase in non-classical correlations follows directly from the increase in Retrieval Efficiency seen in Fig. 5. Error bars are calculated assuming Poissonian statistics.

Equations (4)

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I O U T ( z ) I I N = e a ( δ , T , P ) z
a ( T ) = f ( T ) exp [ 4 ln 2 δ i 2 ( δ v D ( T ) ) 2 ]
a ( T , P ) = f ( T ) α P π ( δ i β P ) 2 ( α P 2 ) 2
g A , B ( 2 ) = p A B p A p B = R E p B

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