Abstract

A theoretical model for calculation of the intrinsic linewidth of QCLs is built on the basis of the quantum Langevin approach. It differs from the traditional rate equation model in that the resonant tunneling and the dynamics of coherent interaction can be considered. Results show that the coupling strength and the dephasing rate associated with resonant tunneling strongly affect the linewidth of THz QCLs in the incoherent resonant-tunneling transport regime but only induce little influence in the coherent regime. The dynamics of coherent interaction and resonant-tunneling transport show insignificant effects on the linewidth calculation of mid-infrared QCLs due to strong coupling in resonant tunneling. We also demonstrate that by properly designing the active regions of QCLs, one can reduce the intrinsic linewidth according to our model.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012 (1)

2011 (2)

2010 (3)

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18(25), 25922–25927 (2010).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

2009 (2)

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[CrossRef]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[CrossRef]

2008 (2)

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

2006 (1)

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

2005 (1)

H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98(10), 104505 (2005).
[CrossRef]

2001 (1)

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001).
[CrossRef]

1998 (1)

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

1996 (2)

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68(1), 127–173 (1996).
[CrossRef]

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68(3), 801–853 (1996).
[CrossRef]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

1993 (1)

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5(6), 640–642 (1993).
[CrossRef]

1983 (1)

B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983).
[CrossRef]

1982 (1)

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[CrossRef]

Aellen, T.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5(6), 640–642 (1993).
[CrossRef]

Akikusa, N.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

Andrews, A. M.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Bartalini, S.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Beck, M.

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001).
[CrossRef]

Benz, A.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Blaser, S.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

Borri, S.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Bowden, C. M.

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5(6), 640–642 (1993).
[CrossRef]

Callebaut, H.

H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98(10), 104505 (2005).
[CrossRef]

Cancio, P.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Capasso, F.

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Capsso, F.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Castrillo, A.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Cho, A. Y.

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Curl, R. F.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Daino, B.

B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983).
[CrossRef]

Davidovich, L.

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68(1), 127–173 (1996).
[CrossRef]

De Natale, P.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Deutsch, C.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Di Francesco, J.

Edamura, T.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

Faist, J.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001).
[CrossRef]

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Fasching, G.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Fujita, K.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

Galli, I.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Gianfrani, L.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Giovannini, M.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

Giusfredi, G.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Gmachl, C.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Henry, C.

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[CrossRef]

Henry, C. H.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68(3), 801–853 (1996).
[CrossRef]

Hofstetter, D.

L. Tombez, S. Schilt, J. Di Francesco, P. Thomann, and D. Hofstetter, “Temperature dependence of the frequency noise in a mid-IR DFB quantum cascade laser from cryogenic to room temperature,” Opt. Express 20(7), 6851–6859 (2012).
[CrossRef] [PubMed]

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001).
[CrossRef]

Hoyler, N.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

Hu, Q.

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[CrossRef]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[CrossRef]

H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98(10), 104505 (2005).
[CrossRef]

Hutchinson, A. L.

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Hvozdara, L.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

Jirauschek, C.

Kan, H.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

Kazarinov, R. F.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68(3), 801–853 (1996).
[CrossRef]

Klang, P.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Kosterev, A. A.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Kumar, S.

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[CrossRef]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[CrossRef]

Lewicki, R.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Liu, T.

T. Liu and Q. J. Wang, “Fundamental frequency noise and linewidth broadening caused by intrinsic temperature fluctuations in quantum cascade lasers,” Phys. Rev. B 84(12), 125322 (2011).
[CrossRef]

Maulini, R.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

Mazzotti, D.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

McManus, B.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Piazzolla, S.

B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983).
[CrossRef]

Pusharsky, M.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Reno, J. L.

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[CrossRef]

Schilt, S.

Schrenk, W.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Sirtori, C.

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Spano, P.

B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983).
[CrossRef]

Strasser, G.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Tamburrini, M.

B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983).
[CrossRef]

Terazzi, R.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

Thomann, P.

Tittel, F. K.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Tombez, L.

Unterrainer, K.

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Wang, Q. J.

T. Liu and Q. J. Wang, “Fundamental frequency noise and linewidth broadening caused by intrinsic temperature fluctuations in quantum cascade lasers,” Phys. Rev. B 84(12), 125322 (2011).
[CrossRef]

Wysocki, G.

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

Yamanishi, M.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

Appl. Phys. Lett. (3)

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett. 89(9), 091121 (2006).
[CrossRef]

S. Kumar, Q. Hu, and J. L. Reno, “186 K operation of terahertz quantum-cascade lasers based on a diagonal design,” Appl. Phys. Lett. 94(13), 131105 (2009).
[CrossRef]

D. Hofstetter, M. Beck, T. Aellen, and J. Faist, “High-temperature operation of distributed feedback quantum-cascade lasers at 5.3 μm,” Appl. Phys. Lett. 78(4), 396 (2001).
[CrossRef]

Chem. Phys. Lett. (1)

R. F. Curl, F. Capsso, C. Gmachl, A. A. Kosterev, B. McManus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. 487(1-3), 1–18 (2010).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44(1), 12–29 (2008).
[CrossRef]

C. Sirtori, F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Resonant tunneling in quantum cascade lasers,” IEEE J. Quantum Electron. 34(9), 1722–1729 (1998).
[CrossRef]

B. Daino, P. Spano, M. Tamburrini, and S. Piazzolla, “Phase noise and spectral line shape in semiconductor lasers,” IEEE J. Quantum Electron. 19(3), 266–270 (1983).
[CrossRef]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5(6), 640–642 (1993).
[CrossRef]

J. Appl. Phys. (1)

H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98(10), 104505 (2005).
[CrossRef]

Mater. Sci. Eng. B (1)

A. M. Andrews, A. Benz, C. Deutsch, G. Fasching, K. Unterrainer, P. Klang, W. Schrenk, and G. Strasser, “Doping dependence of LO-phonon depletion scheme THz quantum-cascade lasers,” Mater. Sci. Eng. B 147(2-3), 152–155 (2008).
[CrossRef]

Opt. Express (3)

Phys. Rev. B (2)

T. Liu and Q. J. Wang, “Fundamental frequency noise and linewidth broadening caused by intrinsic temperature fluctuations in quantum cascade lasers,” Phys. Rev. B 84(12), 125322 (2011).
[CrossRef]

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B 80(24), 245316 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104(8), 083904 (2010).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68(1), 127–173 (1996).
[CrossRef]

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68(3), 801–853 (1996).
[CrossRef]

Science (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Other (4)

M. S. Vitiello, L. Consolino, S. Bartalini, A. Tredicucci, M. Inguscio, and P. De Natale, “The intrinsic linewidth of THz quantum cascade lasers,” CLEO: Science and Innovations, (Optical Society of America, 2012), paper CTu2B.2.

M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University Press, 1997).

W. H. Louisell, Quantum statistical properties of radiation (Wiley, 1973).

K. Petermann, Laser diode modulation and noise (Kluwar Academic, 1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram showing the conduction-band of a three-level QCL structure and the square modulus of the wave functions. Levels 1′, 2 and 3 are the injector level, the lower laser level and the upper laser level, respectively. The inset showing the simplified energy levels with resonant tunneling and optical response. Δ 0 /2 denotes the interaction of the resonant tunneling, g is the electron-light interaction.

Fig. 2
Fig. 2

The comparisons of linewidths derived from our model and the classical rate equation model at different operation currents (a) in THz QCLs; The default parameters are chosen from the typical values for resonant-tunneling injection THz QCLs designs (Refs. 10,23): γ23 = 4.5 × 1012 s−1, γ31′ = 2.5 × 1012 s−1, κ = 2 × 1011 s−1, 1/τ32 = 3 × 1011 s−1, γ3 = 5 × 1011 s−1, γ2 = 2 × 1012 s−1, ε31′ = 0, ∆0 = 1.5 meV, μ = 3.7 nm, Γ ≈1, n = 3.6, T = 20 K, the doping sheet density is 3 × 1010 cm−2. (b) in Mid-IR QCLs; The default parameters are chosen from two phonon resonance mid-IR QCLs designs (Ref. 24): γ23 = 20 × 1012 s−1, κ = 2 × 1011 s−1, 1/τ32 = 3 × 1011 s−1, γ3 = 8 × 1011 s−1, γ2 = 4 × 1012 s−1, γ31′ = 15 × 1012 s−1, ε31′ = 0, ∆0 = 8 meV, μ = 2.1 nm, Γ ≈0.5, n = 3.2, T = 300 K, the doping sheet density is 2 × 1011 cm−2.

Fig. 3
Fig. 3

The effects of coupling strength on the linewidth of THz QCLs at resonance. (a) Linewidth as a function of coupling strength. The linewidth decreases as the coupling strength between the injector level and the upper laser level increases. The factor ( Δ 0 / ) 2 ( γ 3 1 γ 3 ) 1 is used to determine the transition of resonant tunneling from coherent ( 1 ) to incoherent ( 1 ) (Ref. 10). Coupling strengths more than 3 meV lay in the coherent regime. (b) Current density and photon number as a function of coupling strength.

Fig. 4
Fig. 4

The effects of dephasing rate associated with resonant tunneling on the linewidth of THz QCLs at resonance. (a) Linewidth as a function of dephasing rate. The linewidth increases as the dephasing rate increases. (b) Current density and photon number as a function of dephasing rate.

Fig. 5
Fig. 5

The effects of coupling strength on the linewidth of mid-IR QCLs at resonance. (a) Linewidth as a function of coupling strength. The linewidth slowly decreases as the coupling strength of the injector level and the upper laser level increases. (b) Current density and photon number as a function of coupling strength.

Fig. 6
Fig. 6

The effects of dephasing rate associated with resonant tunneling on the linewidth of mid-IR QCLs at resonance. (a) Linewidth as a function of dephasing rate. The linewidth slowly increases as the dephasing rate increases. (b) Current density and photon number as a function of dephasing rate.

Fig. 7
Fig. 7

Effect of doping density on the intrinsic linewidth of THz QCLs at resonance. (a) The self-induced linewidth variation by doping, the linewidth variation due to the change of doping induced cavity loss, and the overall effects from these two factors. (b) Current density and photon number as a function of doping density.

Fig. 8
Fig. 8

(a) Linewidth as a function of relaxation rate of the upper laser level. (b) Linewidth as a function of the relaxation rate of the lower laser level

Fig. 9
Fig. 9

(a) Ratio of current to threshold current and photon number as a function of relaxation rate of the upper laser level. (b) Ratio of current to threshold current and photon number as a function of relaxation rate of the lower laser level.

Equations (78)

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H= ω λ c c+ j ε j b j b j +( g * a b 2 b 3 +g b 3 b 2 a ) ( Δ 0 /2 )( b 3 b 1 + b 1 b 3 )
g= eμ ω λ Γ 2 ε 0 n 2 V m
da dt = κ 2 ai g * σ 23 + f a ( t )
d σ 23 dt = γ 23 σ 23 +ig( ρ 3 ρ 2 )a+ f 23 ( t )
d σ 3 1 dt = γ 3 1 σ 3 1 i Δ 3 1 σ 3 1 i Δ 0 2 ( ρ 1 ρ 3 )+ f 3 1 ( t )
d ρ 3 dt = γ sp ρ 3 γ 3 ρ 3 +i( g * a σ 23 g σ 23 a ) i Δ 0 2 ( σ 3 1 σ 3 1 )+ f 3 ( t )
d ρ 2 dt = γ sp ρ 3 +( 1/ τ 32 ) ρ 3 γ 2 ρ 2 +i( g σ 23 a g * a σ 23 )+ f 2 ( t )
f a (t) =0
f a (t) f a ( t ) =κ n th δ( t t )
f a (t) f a ( t ) =κ( n th +1 )δ( t t )
f a (t) f a ( t ) = f a (t) f a ( t ) =0
n th = [ exp( ω λ / k B T e )1 ] 1
dA dt = κ 2 Ai g * p 23 + F A ( t )
d p 23 dt = γ 23 p 23 +ig( n 3 n 2 )A+ F 23 ( t )
d p 3 1 dt = γ 3 1 p 3 1 i Δ 3 1 p 3 1 i Δ 0 2 ( n 1 n 3 )+ F 3 1 ( t )
d n 3 dt = γ sp n 3 γ 3 n 3 +i( g * A p 23 g p 23 A ) i Δ 0 2 ( p 3 1 p 3 1 )+ F 3 ( t )
d n 2 dt = γ sp n 3 +( 1/ τ 32 ) n 3 γ 2 n 2 +i( g p 23 A g * A p 23 )+ F 2 ( t )
F μ (t) =0
F u (t) F v ( t ) =2 D u v δ( t t )
I 0 = A 0 2 = γ 2 n 2,0 γ sp n 3,0 ( 1/ τ 32 ) n 3,0 κ
n 3,0 n 2,0 = γ 23 κ 2g g *
n 3,0 = [ γ 2 γ 23 κ 2g g * ( Δ 0 2 ) 2 ( N 0 + γ 23 κ 2g g * ) 2 γ 3 1 ( γ 3 1 ) 2 + ( Δ 3 1 / ) 2 ] / [ 1/ τ 32 γ 3 γ 2 ( Δ 0 2 ) 2 6 γ 3 1 γ 3 1 2 + ( Δ 3 1 / ) 2 ]
n 2,0 = [ γ 23 (1/ τ 32 γ 3 )κ 2g g * + ( Δ 0 2 ) 2 ( γ 23 κ g g * N 0 ) 2 γ 3 1 γ 3 1 2 + ( Δ 3 1 / ) 2 ] / [ 1/ τ 32 γ 3 γ 2 ( Δ 0 2 ) 2 6 γ 3 1 γ 3 1 2 + ( Δ 3 1 / ) 2 ]
p 23,0 = κ i2g A 0
p 3 1 ,0 = i Δ 0 2 ( N 0 n 2,0 2 n 3,0 ) / ( γ 3 1 +i Δ 1 3 )
I c,th =| e |( n 3,th τ 31 + n 2,th τ 21 )=| e | γ 23 κ γ 2 /2 τ 31 +( γ sp +1/ τ 32 )/2 τ 21 ( γ 2 γ sp 1/ τ 32 )g g *
I c =| e |( n 3,0 τ 31 + n 2,0 τ 21 )=| e | ( Δ 0 2 ) 2 [ ( τ 31 + τ 21 ) N 0 τ 31 τ 21 + (2 τ 31 τ 21 ) γ 23 κ 2 τ 31 τ 21 g g * ] 2 γ 3 1 ( γ 3 1 ) 2 + ( Δ 3 1 / ) 2 1/ τ 32 γ 3 γ 2 ( Δ 0 2 ) 2 6 γ 3 1 γ 3 1 2 + ( Δ 3 1 / ) 2
A= A 0 +δA
p 23 = p 23,0 +δ p 23
p 3 1 = p 3 1 ,0 +δ p 3 1
n 3 = n 3,0 +δ n 3
n 2 = n 2,0 +δ n 2
dδA dt = κ 2 δAi g * δ p 23 + F A ( t )
dδ p 23 dt = γ 23 δ p 23 +ig(δ n 3 δ n 2 ) A 0 +ig( n 3,0 n 2,0 )δA+ F 23 ( t )
dδ p 3 1 dt = γ 3 1 δ p 3 1 i Δ 3 1 δ p 3 1 i Δ 0 2 (δ n 1 δ n 3 )+ F 3 1 ( t )
dδ n 3 dt = γ sp δ n 3 γ 3 δ n 3 +i( g * δ A p 23,0 g p 23,0 δA )+i( g * A 0 δ p 23 gδ p 23 A 0 ) i Δ 0 2 ( δ p 3 1 δ p 3 1 )+ F 3 ( t )
dδ n 2 dt = γ sp δ n 3 +( 1/ τ 32 )δ n 3 γ 2 δ n 2 +i( g p 23,0 δA g * δ A p 23,0 )+i( gδ p 23 A 0 g * A 0 δ p 23 )+ F 2 ( t )
δ n 3 ( ω )= + e iωt δ n 3 ( t )dt
iωδA( ω )=( κ/2 )δA( ω )i g * δ p 23 ( ω )+ F A ( ω )
iωδ p 23 ( ω )= γ 23 δ p 23 ( ω )+ig[ δ n 3 ( ω )δ n 2 ( ω ) ] A 0 +ig( n 3,0 n 2,0 )δA( ω )+ F 23 ( ω )
iωδ p 3 1 ( ω )= γ 3 1 δ p 3 1 ( ω )i Δ 1 3 δ p 3 1 ( ω ) i Δ 0 2 [ δ n 1 ( ω )δ n 3 ( ω ) ]+ F 3 1 ( ω )
iωδ n 3 ( ω )= γ sp δ n 3 ( ω ) γ 3 δ n 3 ( ω )+i[ g * δ A ( ω ) p 23,0 g p 23,0 δA( ω ) ] +i[ g * A 0 δ p 23 ( ω )gδ p 23 ( ω ) A 0 ] i Δ 0 2 [ δ p 3 1 ( ω )δ p 3 1 ( ω ) ]+ F 3 ( ω )
iωδ n 2 ( ω )= γ sp δ n 3 ( ω )+( 1/ τ 32 )δ n 3 ( ω ) γ 2 δ n 2 ( ω ) +i[ g p 23,0 δA( ω ) g * δ A ( ω ) p 23,0 ]+i[ gδ p 23 ( ω ) A 0 g * A 0 δ p 23 ( ω ) ]+ F 2 ( ω )
F u ( ω ) F v ( ω ) =4π D uv δ( ω+ ω )
A= I exp( iφ )
δφ( t )= i 2 I 0 [δ A ( t )δA( t )]
δφ( ω )= i 2 I 0 [δ A ( ω )δA( ω )]
δA( ω )= igδ p 23 ( ω )+ F A ( ω ) iω+κ/2
δ p 23 ( ω )= ig[ δ n 3 ( ω )δ n 2 ( ω ) ] A 0 +ig( n 3,0 n 2,0 ) F A ( ω ) iω+κ/2 + F 23 ( ω ) iω+ γ 23 g g * ( n 3,0 n 2,0 ) iω+κ/2
δφ( ω )= i 2 I 0 ( iω+ γ 23 )[ F A (ω) F A (ω) ]+ig[ F 23 ( ω ) F 23 ( ω ) ] ( iω+ γ 23 )( iω+κ/2 )g g * ( n 3,0 n 2,0 )
δφ( ω )δ φ * ( ω ) = ( δφ ) ω 2 δ( ω ω )
δφ(ω)δ φ * ( ω ) =2π S φ δ( ω ω )
S φ = + δφ(t)δφ(tτ) exp( iωτ ) dτ
S f = ω 2 S φ = κ( ω 2 + γ 23 2 )( 2 n th +1 )+2g g * γ 23 ( n 3,0 + n 2,0 2 n 3,0 n 2,0 / N 0 ) 4 I 0 [ ω 2 + ( γ 23 +κ/2 ) 2 ]
A(t) A * (tτ) = I 0 exp( i ω λ t )exp( δ φ 2 /2 )
δ φ 2 = 2 π 0 S f ( ω ) ω 2 [ 1cos( ωt ) ] dω
δ φ 2 = 2κ n th +κ+2( g g * / γ 23 )( n 3,0 + n 2,0 2 n 3,0 n 2,0 / N 0 ) 4 ( 1+κ/ 2 γ 23 ) 2 I 0 t + 2 κ 2 n th / γ 23 κ 2 / γ 23 +2( g g * / γ 23 )( n 3,0 + n 2,0 2 n 3,0 n 2,0 / N 0 ) 4 ( 1+κ/ 2 γ 23 ) 2 I 0 [ e ( γ 23 +κ/2 )| t | 1 γ 23 +κ/2 ]
S A ( ω )= I 0 + exp[ i(ω ω λ )t ] exp( δ φ 2 /2 )dτ
δf= κ(2 n th +1)+2( g g * / γ 23 ) N 0 n 3,0 8π (1+κ/ 2 γ 23 ) 2 I 0
d o u dt = d u + f u (t)
f u (t) =0
f u (t) f v ( t ) =2 d u v δ(t t )
2 d u v = d dt o u o v d u o v o u d v
f 3 (t) f 3 ( t ) =2 d 33 δ(t t )
f 3 (t) f 3 ( t ) = d dt ( ρ 3 ) 2 ( d dt ρ 3 ) ρ 3 ρ 3 ( d dt ρ 3 ) δ(t t ) = γ sp ρ 3 γ 3 ρ 3 +i( g * a σ 23 g σ 23 a) i Δ 0 2 ( σ 3 1 σ 3 1 ) γ sp ρ 3 ρ 3 γ 3 ρ 3 ρ 3 +i( g * a σ 23 g σ 23 a) ρ 3 i Δ 0 2 ( σ 3 1 σ 3 1 ) ρ 3 γ sp ρ 3 ρ 3 γ 3 ρ 3 ρ 3 +i ρ 3 ( g * a σ 23 g σ 23 a) i Δ 0 2 n 3 ( σ 3 1 σ 3 1 ) = ( γ sp + γ 3 ) ρ 3 δ(t t )
f 2 (t) f 2 ( t ) = ( γ sp +1/ τ 32 ) ρ 3 + γ 2 ρ 2 δ(t t )
f 23 (t) f 23 ( t ) = (2 γ 23 γ sp 1 / τ 32 ) ρ 3 + γ 2 ρ 2 δ(t t )
f 23 (t) f 23 ( t ) = ( γ sp +1 / τ 32 ) ρ 3 +(2 γ 23 γ 2 ) ρ 2 δ(t t )
f 23 (t) f 23 ( t ) =0
f 3 1 (t) f 3 1 ( t ) = γ 2 n 2 +( γ 3 1/ τ 32 ) ρ 3 +2 γ 3 1 ρ 1 δ(t t )
f 3 1 (t) f 3 1 ( t ) = (1 / τ 32 γ 3 +2 γ 3 1 ) ρ 3 γ 2 ρ 2 δ(t t )
d O u dt = D u + F u (t)
2 D u v = d dt O u O v D u O v O u D v
d dt o u o v = d dt O u O v
F u F v = f u f v + d u o v + o u d v D u O v O u D v
F A (t) F A ( t ) = f a (t) f a ( t )
F 23 (t) F 23 ( t ) = f 23 (t) f 23 ( t )
F 23 (t) F 23 ( t ) = f 23 (t) f 23 ( t ) +2ig p 23 A δ(t t )

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