Abstract

We demonstrate transient saturation spectroscopy for Er3+167 ions doped in a silicate glass fiber cooled at 2.530K to measure the population relaxation time t1 of the hyperfine sublevels. The observed t1 value is 3.1ms at 4K and we observe anomalous temperature dependence whereby t1 becomes rather longer with heating from 4 to 30K. We can regard the population relaxation as a result of the Raman scattering of the Boson peak mode (BPM) peculiar to a silicate glass by the 4f-electrons of the Er3+167 ions. We can attribute the anomalous temperature dependence to the suppression of the Raman scattering by thermal hopping of the localized BPM.

© 2011 Optical Society of America

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  1. R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995).
    [CrossRef]
  2. R. Yano, M. Mitsunaga, and N. Uesugi, “Ultralong optical dephasing time in Eu3+:Y2SiO5,” Opt. Lett. 16, 1884–1886 (1991).
    [CrossRef] [PubMed]
  3. R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, “Ultraslow optical dephasing in Eu3+:Y2SiO5,” Phys. Rev. Lett. 72, 2179–2182 (1994).
    [CrossRef] [PubMed]
  4. X. A. Shen and R. Kachru, “F07-D15 transition in Eu3+:Y2SiO5,” J. Opt. Soc. Am. B 11, 591–596 (1994).
    [CrossRef]
  5. S. B. Altner, G. Zumofen, U. P. Wild, and M. Mitsunaga, “Photon-echo attenuation in rare-earth-ion-doped crystals,” Phys. Rev. B 54, 17493–17507 (1996).
    [CrossRef]
  6. K. Holliday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SiO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993).
    [CrossRef]
  7. F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
    [CrossRef]
  8. T. Blasberg and D. Suter, “Bichromatic excitation of coherent Raman beats in rare-earth solids,” Phys. Rev. B 51, 6309–6318(1995).
    [CrossRef]
  9. H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
    [CrossRef] [PubMed]
  10. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
    [CrossRef]
  11. R. M. Macfarlane and R. M. Shelby, “Measurement of optical dephasing of Eu3+ and Pr3+ doped silicate glasses by spectral hole burning,” Opt. Commun. 45, 46–51 (1983).
    [CrossRef]
  12. R. M. Macfarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
    [CrossRef]
  13. R. Wannemacher, J. M. A. Koedijk, and S. Völker, “Dynamics of spectral holes in rare-earth-doped glass fibers,” J. Lumin. 60–61, 437–440 (1994).
    [CrossRef]
  14. E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
    [CrossRef] [PubMed]
  15. M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
    [CrossRef] [PubMed]
  16. B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
    [CrossRef] [PubMed]
  17. T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
    [CrossRef]
  18. T. Böttger, Y. Sun, C. W. Thiel, and R. L. Cone, “Spectroscopy and dynamics of Er3+:Y2SiO5 at 1.5 μm,” Phys. Rev. B 74, 075107 (2006).
    [CrossRef]
  19. E. Baldit, S. Briaudeau, P. Monnier, J. A. Levenson, and K. Bencheikh, “Electromagnetically induced transparency in rare-earth Er3+167-doped Y2SiO5 crystal,” in Quantum Electronics and Laser Science Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper QTuE7.
  20. R. M. Macfarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602(2006).
    [CrossRef] [PubMed]
  21. M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
    [CrossRef]
  22. Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
    [CrossRef] [PubMed]
  23. T. Nakayama, “Boson peak and terahertz frequency dynamics of vitreous silica,” Rep. Prog. Phys. 65, 1195–1242 (2002).
    [CrossRef]
  24. J. H. Van Vleck, “Paramagnetic relaxation times for titanium and chrome alum,” Phys. Rev. 57, 426–447 (1940).
    [CrossRef]
  25. D. Hashimoto and K. Shimizu, “Cooling an optical fiber to 4.5 K by indirect thermal contact with a liquid-helium flow and spectroscopic temperature measurements,” Rev. Sci. Instrum. 79, 093102 (2008).
    [CrossRef] [PubMed]
  26. T. Nakayama, “Low-energy excitations in water: a simple-model analysis,” Phys. Rev. Lett. 80, 1244–1247 (1998).
    [CrossRef]
  27. M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
    [CrossRef]
  28. R. C. Zeller and R. O. Pohl, “Thermal conductivity and specific heat of noncrystalline solids,” Phys. Rev. B 4, 2029–2041(1971).
    [CrossRef]
  29. T. Nakayama and R. Orbach, “Anharmonicity and thermal transport in network glasses,” Europhys. Lett. 47, 468–473 (1999).
    [CrossRef]
  30. V. L. Gurevich, D. A. Parshin, and H. R. Schober, “Anharmonicity, vibrational instability, and the Boson peak in glasses,” Phys. Rev. B 67, 094203 (2003).
    [CrossRef]
  31. L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
    [CrossRef]
  32. A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
    [CrossRef] [PubMed]

2011

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
[CrossRef] [PubMed]

2010

B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
[CrossRef] [PubMed]

2009

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
[CrossRef]

2008

H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
[CrossRef] [PubMed]

D. Hashimoto and K. Shimizu, “Cooling an optical fiber to 4.5 K by indirect thermal contact with a liquid-helium flow and spectroscopic temperature measurements,” Rev. Sci. Instrum. 79, 093102 (2008).
[CrossRef] [PubMed]

2007

M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
[CrossRef] [PubMed]

2006

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef] [PubMed]

T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
[CrossRef]

T. Böttger, Y. Sun, C. W. Thiel, and R. L. Cone, “Spectroscopy and dynamics of Er3+:Y2SiO5 at 1.5 μm,” Phys. Rev. B 74, 075107 (2006).
[CrossRef]

R. M. Macfarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602(2006).
[CrossRef] [PubMed]

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

2005

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
[CrossRef] [PubMed]

2003

F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
[CrossRef]

V. L. Gurevich, D. A. Parshin, and H. R. Schober, “Anharmonicity, vibrational instability, and the Boson peak in glasses,” Phys. Rev. B 67, 094203 (2003).
[CrossRef]

2002

L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

T. Nakayama, “Boson peak and terahertz frequency dynamics of vitreous silica,” Rep. Prog. Phys. 65, 1195–1242 (2002).
[CrossRef]

2001

M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
[CrossRef]

1999

T. Nakayama and R. Orbach, “Anharmonicity and thermal transport in network glasses,” Europhys. Lett. 47, 468–473 (1999).
[CrossRef]

1998

T. Nakayama, “Low-energy excitations in water: a simple-model analysis,” Phys. Rev. Lett. 80, 1244–1247 (1998).
[CrossRef]

1996

S. B. Altner, G. Zumofen, U. P. Wild, and M. Mitsunaga, “Photon-echo attenuation in rare-earth-ion-doped crystals,” Phys. Rev. B 54, 17493–17507 (1996).
[CrossRef]

1995

T. Blasberg and D. Suter, “Bichromatic excitation of coherent Raman beats in rare-earth solids,” Phys. Rev. B 51, 6309–6318(1995).
[CrossRef]

R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995).
[CrossRef]

1994

R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, “Ultraslow optical dephasing in Eu3+:Y2SiO5,” Phys. Rev. Lett. 72, 2179–2182 (1994).
[CrossRef] [PubMed]

X. A. Shen and R. Kachru, “F07-D15 transition in Eu3+:Y2SiO5,” J. Opt. Soc. Am. B 11, 591–596 (1994).
[CrossRef]

R. Wannemacher, J. M. A. Koedijk, and S. Völker, “Dynamics of spectral holes in rare-earth-doped glass fibers,” J. Lumin. 60–61, 437–440 (1994).
[CrossRef]

1993

K. Holliday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SiO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993).
[CrossRef]

1991

1987

R. M. Macfarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
[CrossRef]

1983

R. M. Macfarlane and R. M. Shelby, “Measurement of optical dephasing of Eu3+ and Pr3+ doped silicate glasses by spectral hole burning,” Opt. Commun. 45, 46–51 (1983).
[CrossRef]

1971

R. C. Zeller and R. O. Pohl, “Thermal conductivity and specific heat of noncrystalline solids,” Phys. Rev. B 4, 2029–2041(1971).
[CrossRef]

1940

J. H. Van Vleck, “Paramagnetic relaxation times for titanium and chrome alum,” Phys. Rev. 57, 426–447 (1940).
[CrossRef]

Afzelius, M.

B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
[CrossRef] [PubMed]

H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
[CrossRef] [PubMed]

M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
[CrossRef] [PubMed]

M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
[CrossRef]

Altner, S. B.

S. B. Altner, G. Zumofen, U. P. Wild, and M. Mitsunaga, “Photon-echo attenuation in rare-earth-ion-doped crystals,” Phys. Rev. B 54, 17493–17507 (1996).
[CrossRef]

Arai, M.

M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
[CrossRef]

Baldit, E.

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
[CrossRef] [PubMed]

E. Baldit, S. Briaudeau, P. Monnier, J. A. Levenson, and K. Bencheikh, “Electromagnetically induced transparency in rare-earth Er3+167-doped Y2SiO5 crystal,” in Quantum Electronics and Laser Science Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper QTuE7.

Balogh, J.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
[CrossRef] [PubMed]

Bayart, D.

L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

Bencheikh, K.

E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
[CrossRef] [PubMed]

E. Baldit, S. Briaudeau, P. Monnier, J. A. Levenson, and K. Bencheikh, “Electromagnetically induced transparency in rare-earth Er3+167-doped Y2SiO5 crystal,” in Quantum Electronics and Laser Science Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper QTuE7.

Bennington, S. M.

M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
[CrossRef]

Bigot, L.

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef] [PubMed]

L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

Blasberg, T.

T. Blasberg and D. Suter, “Bichromatic excitation of coherent Raman beats in rare-earth solids,” Phys. Rev. B 51, 6309–6318(1995).
[CrossRef]

Bosak, A.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
[CrossRef] [PubMed]

Böttger, T.

T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
[CrossRef]

T. Böttger, Y. Sun, C. W. Thiel, and R. L. Cone, “Spectroscopy and dynamics of Er3+:Y2SiO5 at 1.5 μm,” Phys. Rev. B 74, 075107 (2006).
[CrossRef]

Briaudeau, S.

E. Baldit, S. Briaudeau, P. Monnier, J. A. Levenson, and K. Bencheikh, “Electromagnetically induced transparency in rare-earth Er3+167-doped Y2SiO5 crystal,” in Quantum Electronics and Laser Science Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper QTuE7.

Chumakov, A. I.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
[CrossRef] [PubMed]

Comez, L.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
[CrossRef] [PubMed]

Cone, R. L.

T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
[CrossRef]

T. Böttger, Y. Sun, C. W. Thiel, and R. L. Cone, “Spectroscopy and dynamics of Er3+:Y2SiO5 at 1.5 μm,” Phys. Rev. B 74, 075107 (2006).
[CrossRef]

R. M. Macfarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602(2006).
[CrossRef] [PubMed]

Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef] [PubMed]

F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
[CrossRef]

R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995).
[CrossRef]

R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, “Ultraslow optical dephasing in Eu3+:Y2SiO5,” Phys. Rev. Lett. 72, 2179–2182 (1994).
[CrossRef] [PubMed]

Crichton, W. A.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
[CrossRef] [PubMed]

Croci, M.

K. Holliday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SiO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993).
[CrossRef]

de Riedmatten, H.

B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
[CrossRef] [PubMed]

H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
[CrossRef] [PubMed]

M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
[CrossRef] [PubMed]

Equall, R. W.

F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
[CrossRef]

R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995).
[CrossRef]

R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, “Ultraslow optical dephasing in Eu3+:Y2SiO5,” Phys. Rev. Lett. 72, 2179–2182 (1994).
[CrossRef] [PubMed]

Fioretto, D.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
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H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
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M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
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M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
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V. L. Gurevich, D. A. Parshin, and H. R. Schober, “Anharmonicity, vibrational instability, and the Boson peak in glasses,” Phys. Rev. B 67, 094203 (2003).
[CrossRef]

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D. Hashimoto and K. Shimizu, “Cooling an optical fiber to 4.5 K by indirect thermal contact with a liquid-helium flow and spectroscopic temperature measurements,” Rev. Sci. Instrum. 79, 093102 (2008).
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M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
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M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
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K. Holliday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SiO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
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M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
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M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
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Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
[CrossRef] [PubMed]

L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
[CrossRef]

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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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L. Bigot, A. Jurdyc, B. Jacquier, L. Gasca, and D. Bayart, “Resonant fluorescence line narrowing measurements in erbium-doped glass for optical amplifiers,” Phys. Rev. B 66, 214204 (2002).
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Kargl, F.

A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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R. Wannemacher, J. M. A. Koedijk, and S. Völker, “Dynamics of spectral holes in rare-earth-doped glass fibers,” J. Lumin. 60–61, 437–440 (1994).
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F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
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B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
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E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
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E. Baldit, S. Briaudeau, P. Monnier, J. A. Levenson, and K. Bencheikh, “Electromagnetically induced transparency in rare-earth Er3+167-doped Y2SiO5 crystal,” in Quantum Electronics and Laser Science Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper QTuE7.

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A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
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R. M. Macfarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602(2006).
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F. Könz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003).
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R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, “Ultraslow optical dephasing in Eu3+:Y2SiO5,” Phys. Rev. Lett. 72, 2179–2182 (1994).
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R. M. Macfarlane and R. M. Shelby, “Measurement of optical dephasing of Eu3+ and Pr3+ doped silicate glasses by spectral hole burning,” Opt. Commun. 45, 46–51 (1983).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
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S. B. Altner, G. Zumofen, U. P. Wild, and M. Mitsunaga, “Photon-echo attenuation in rare-earth-ion-doped crystals,” Phys. Rev. B 54, 17493–17507 (1996).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
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E. Baldit, S. Briaudeau, P. Monnier, J. A. Levenson, and K. Bencheikh, “Electromagnetically induced transparency in rare-earth Er3+167-doped Y2SiO5 crystal,” in Quantum Electronics and Laser Science Conference, 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper QTuE7.

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M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
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M. Nakamura, M. Arai, T. Otomo, Y. Inamura, and S. M. Bennington, “Dispersive excitation in different forms of SiO2,” J. Non-Cryst. Solids 293–295, 377–382 (2001).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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V. L. Gurevich, D. A. Parshin, and H. R. Schober, “Anharmonicity, vibrational instability, and the Boson peak in glasses,” Phys. Rev. B 67, 094203 (2003).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. 95, 143601 (2005).
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A. I. Chumakov, G. Monaco, A. Monaco, W. A. Crichton, A. Bosak, R. Rüffer, A. Meyer, F. Kargl, L. Comez, D. Fioretto, H. Giefers, S. Roitsch, G. Wortmann, M. H. Manghnani, A. Hushur, Q. Williams, J. Balogh, K. Parliński, P. Jochym, and P. Piekarz, “Equivalence of the Boson peak in glasses to the transverse acoustic van Hove singularity in crystals,” Phys. Rev. Lett. 106, 225501 (2011).
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A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
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B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
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V. L. Gurevich, D. A. Parshin, and H. R. Schober, “Anharmonicity, vibrational instability, and the Boson peak in glasses,” Phys. Rev. B 67, 094203 (2003).
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R. M. Macfarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602(2006).
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R. M. Macfarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987).
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R. M. Macfarlane and R. M. Shelby, “Measurement of optical dephasing of Eu3+ and Pr3+ doped silicate glasses by spectral hole burning,” Opt. Commun. 45, 46–51 (1983).
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Shimizu, K.

D. Hashimoto and K. Shimizu, “Cooling an optical fiber to 4.5 K by indirect thermal contact with a liquid-helium flow and spectroscopic temperature measurements,” Rev. Sci. Instrum. 79, 093102 (2008).
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B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010).
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H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
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M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
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M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
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H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456, 773–777 (2008).
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M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
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M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
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M. U. Staudt, M. Afzelius, H. De Riedmatten, S. R. Hastings-Simon, C. Simon, R. Ricken, H. Suche, W. Sohler, and N. Gisin, “Interference of multimode photon echoes generated in spatially separated solid-state atomic ensembles,” Phys. Rev. Lett. 99, 173602 (2007).
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Y. Sun, R. L. Cone, L. Bigot, and B. Jacquier, “Exceptionally narrow homogeneous linewidth in erbium-doped glasses,” Opt. Lett. 31, 3453–3455 (2006).
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R. M. Macfarlane, Y. Sun, P. B. Sellin, and R. L. Cone, “Optical decoherence in Er3+-doped silicate fiber: evidence for coupled spin-elastic tunneling systems,” Phys. Rev. Lett. 96, 033602(2006).
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T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
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T. Böttger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μm in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B 73, 075101 (2006).
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A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706–714 (2009).
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M. U. Staudt, S. R. Hastings-Simon, M. Afzelius, D. Jaccard, W. Tittel, and N. Gisin, “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage,” Opt. Commun. 266, 720–726 (2006).
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R. Wannemacher, J. M. A. Koedijk, and S. Völker, “Dynamics of spectral holes in rare-earth-doped glass fibers,” J. Lumin. 60–61, 437–440 (1994).
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Figures (10)

Fig. 1
Fig. 1

Simple three-level model. The population is distributed between these three levels depending on the intensity I 1 of the resonant light to | 1 | 3 optical transition and a set of relaxation factors η 1 , η 2 , and β. After turning off the pump light pulse, the populations of levels | 2 and | 3 relax to level | 1 with relaxation times of t 1 and T 1 , respectively.

Fig. 2
Fig. 2

Simplified energy level structure of Er 167 ions doped in glass. Only Er 167 has a nuclear spin of 7 / 2 , and each Stark level is split again into hyperfine levels. I ˜ is the stimulated absorption and emission rate. About η 1 and η 2 , only the representative transitions are described.

Fig. 3
Fig. 3

(a) Experimental setup. AOM: Acousto-optic modulator. The pump and probe pulses are launched from opposite ends of the 4.5 m sample fiber. The probe pulses transmitted through the sample fiber are detected by the photo detector. (b) Time sequence of the operation: the pump and probe pulse durations are 45 ms , and the period is 100 ms . The probe pulse is launched immediately after the pump pulse is turned off.

Fig. 4
Fig. 4

Temperature dependence of the homogeneous linewidth ν h (FWHM) measured with SHB spectroscopy.

Fig. 5
Fig. 5

Intensity change of the transmitted probe pulse through the sample silicate glass fiber cooled to 4, 10, and 20 K . The averaging numbers for the measurement are 5000, 2000, and 1000, respectively. I S is the saturation intensity at each temperature. The transmitted intensity is normalized with that of the probe pulse at t = 0 and 4 K .

Fig. 6
Fig. 6

Temporal change in the transmitted probe intensity with the components consisting of only Er 3 + 167 ions at 4 K , where a log scale is employed for the vertical axis. From the optimal fitting based on Eq. (A6), the two decay rates λ + and λ are estimated to be 2 π × 99.2 and 2 π × 17.9 Hz , respectively.

Fig. 7
Fig. 7

Temperature dependence of β estimated from the two decay rates λ ± of the transmitted probe intensity.

Fig. 8
Fig. 8

Spectral structure of the Stark levels. Backward fluorescence emitted from the 4.5 m sample fiber cooled to 4 K . The spectrum intensity is normalized with that of the second peak around 1539 nm . The inset is an expanded view of the second and third spectral peaks. From the fitted curve with a double Gaussian function, the spectral width (FWHM) of the second peak is 3.1 nm ( 390 GHz ).

Fig. 9
Fig. 9

Hopping of the BPM to other sites with the assistance of the propagating acoustic phonons when they are significantly thermally excited.

Fig. 10
Fig. 10

Profiles for the T dependence evaluated with Eq. (13). The cross points with the error bars represent the experimental results. The square, circle, and diamond points represent the calculation results of β g , β e , and β g + β e , respectively. In this calculation, parameter values are ( ω max = 2 π × 1 · 10 12 Hz , ω c = 2 π × 5 · 10 9 Hz , C = 2 π × 5 · 10 11 Hz · K 1 ) and | A e ( f , l ) · A e ( l , i ) | 2 0.6 × | A g ( f , l ) · A g ( l , i ) | 2 .

Equations (25)

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d N g A d t = ( N g A N e A ) I ˜ + η 1 N e A 3 β N g A + 3 β N g B ,
d N e A d t = ( N g A N e A ) I ˜ η 1 N e A 3 β N e A + 3 β N e B ,
d N g B d t = η 1 N e B β N g B + β N g A ,
d N e B d t = η 1 N e B β N e B + β N e A .
λ 0 = 4 β ,
λ ± = ( I ˜ 2 + η 1 + 2 β ) ± ( I ˜ 2 + β ) 2 + 3 β 2 .
α ( t ) A 0 B 0 · exp ( λ + t ) C 0 · exp ( λ t ) D 0 · exp [ ( η 1 + 2 I ˜ 2 ) · t ] ,
β = 1 τ = 2 π 2 b p 2 b p 1 1 2 d δ E l i · P l ( δ E l i ) | f , m 1 1 , m 2 + 1 | H e b p | l , m 1 1 , m 2 l , m 1 1 , m 2 | H e b p | i , m 1 , m 2 | 2 [ ( E l i + δ E l i ) / ω b p 1 ] 2 + ( Γ l / 2 + Γ b p 1 ( T ) / 2 ) 2 × δ ( ω b p 1 ω b p 2 E f / + E i / ) .
b p π L b p 3 · ω c d ω b p D b p ( ω b p ) .
D b p ( ω b p ) 1 Ω b p ( Ω b p ω b p ) 4 1 Z + 2 ( ω b p ) + Z 2 ( ω b p ) [ 1 2 Z + ( ω b p ) ln Z + ( ω b p ) + 1 Z + ( ω b p ) 1 + 1 Z ( ω b p ) tan 1 1 Z ( ω b p ) ] ,
Z ± ( ω b p ) = 1 2 9 + 8 ( Ω b p / ω b p ) 6 ± 3 .
Γ b p = 1 τ hop = C [ ( ω ph ( T ) 2 ω b p · ω b p ( R ) ) exp ( 2 R ( T ) L b p ) ] × T .
ω ph ( T ) = ( k B T / ) 2 + ω max 2 ( k B T / ) 4 + ω max 4 .
R ( T ) = ( 3 4 π ) 1 / 3 ( 1 D ¯ b p × ω ph ( T ) ) 1 / 3
f , m 1 1 , m 2 + 1 | H e b p | l , m 1 1 , m 2 = m 2 + 1 2 ω b p · ( 1 L b p ) · A g ( f , l ) ρ · π L b p 3 ,
β g 2 π 2 · 1 4 ρ 2 1 L b p 4 × ω c d ω b p m 1 ( m 2 + 1 ) · D b p ( ω b p ) 2 × δ E l i g · P l g ( δ E l i g ) | A g ( f , l ) · A g ( l , i ) | 2 ω b p 2 [ ( E l i g + δ E l i g ) / ω b p ] 2 + 1 4 ( ω b p Γ l + C ω ph ( T ) 1 + ( ω b p / ω ph ( T ) ) exp [ 2 ( π Δ b p / ω ph ( T ) ) 1 / 3 ] · T ) 2 ,
N g A = η 1 ( η 1 + I ˜ 1 ) + β ( I ˜ 1 + 4 η 1 ) 4 η 1 ( η 1 + 2 I ˜ 1 ) + 8 β ( I ˜ 1 + 2 η 1 ) N ,
N e A = I ˜ 1 ( η 1 + β ) 4 η 1 ( η 1 + 2 I ˜ 1 ) + 8 β ( I ˜ 1 + 2 η 1 ) N ,
N g B = η 1 ( η 1 + 2 I ˜ 1 ) + β ( I ˜ 1 + 4 η 1 ) 4 η 1 ( η 1 + 2 I ˜ 1 ) + 8 β ( I ˜ 1 + 2 η 1 ) N ,
N e B = I ˜ 1 β 4 η 1 ( η 1 + 2 I ˜ 1 ) + 8 β ( I ˜ 1 + 2 η 1 ) N ,
d d t ( f g h ) = ( 4 β 0 0 η 1 ( 2 I ˜ + η 1 + 3 β ) 3 β η 1 / 3 β ( β + η 1 ) ) ( f g h ) + ( β 0 η 1 / 3 ) N .
g ( t ) = A 0 B 0 · exp ( λ + t ) C 0 · exp ( λ t ) .
A 0 = η 1 ( η 1 + 4 β ) 4 η 1 ( η 1 + 2 I ˜ 2 ) + 8 β ( I ˜ 2 + 2 η 1 ) N = N g A ( t f ; I 2 ) N e A ( t f ; I 2 ) ,
B 0 = 1 λ + λ [ ( 3 β + 2 I ˜ 2 + η 1 λ ) g ( 0 ) + 3 β h ( 0 ) λ A + η 1 4 N ] ,
C 0 = 1 λ + λ [ ( 3 β + 2 I ˜ 2 + η 1 λ + ) g ( 0 ) 3 β h ( 0 ) + λ + A η 1 4 N ] .

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