Abstract

Weak material nonlinearity at optical frequencies poses a serious hurdle to realizing optical bistability at low optical powers, which is a critical component for digital optical computing. In this paper, we explore the cavity enhancement of the second-order optical nonlinearity in order to determine the feasibility of few photon optical bistability. Starting from a quantum optical formalism of a doubly resonant cavity (required to meet the condition of phase matching), we derive a dynamic classical model of a cavity that is bistable at the fundamental mode. We analyze the optical energy and the switching speed as a function of the cavity quality factors and mode volumes and identify the regime where only ten’s of photons are required to perform the switching. An unusual trend in the switching speed is also observed, where the speed monotonically decreases as the cavity linewidth increases. This is ascribed to the increase in the switching gain with increasing cavity linewidth.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  4. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nature Photonics 4, 261–263 (2014).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  7. M. Sodagar, M. Miri, A. A. Eftekhar, and A. Adibi, “Optical bistability in a one-dimensional photonic crystal resonator using a reverse-biased pn-junction,” Opt. Express 23, 2676–2685 (2015).
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    [Crossref]
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2015 (1)

2014 (3)

2013 (5)

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7, 597607 (2013).
[Crossref]

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

J. Wang, H. Mabuchi, and X.-L. Qi, “Calculation of divergent photon absorption in ultrathin films of a topological insulator,” Phys. Rev. B 88, 195127 (2013).
[Crossref]

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Physical Review B 87235319 (2013).
[Crossref]

Y.-D. Kwon, M. A. Armen, and H. Mabuchi, “Femtojoule-scale all-optical latching and modulation via cavity nonlinear optics,” Phys. Rev. Lett. 111, 203002 (2013).
[Crossref] [PubMed]

2012 (2)

A. Majumdar, M. Bajcsy, D. Englund, and J. Vuckovic, “All optical switching with a single quantum dot strongly coupled to a photonic crystal cavity,” Selected Topics in Quantum Electronics, IEEE Journal of 18, 1812–1817 (2012).
[Crossref]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

2010 (3)

D. A. B. Miller, “Are optical transistors the logical next step?” Nature Photonics 4, 3–5 (2010).
[Crossref]

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nature Photonics 4, 492–494 (2010).
[Crossref]

2009 (1)

2004 (2)

2000 (1)

X. Wan, J. Dong, M. Qian, and W. Zhang, “Nonlinear optical properties of perovskite ymno3 studied by real-space recursion method,” Phys. Rev. B 61, 10664–10669 (2000).
[Crossref]

1984 (1)

A. Sawchuk and T. C. Strand, “Digital optical computing,” Proceedings of the IEEE 72, 758–779 (1984).
[Crossref]

Adibi, A.

Alencar, T. V.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

Almeida, V. R.

Armen, M. A.

Y.-D. Kwon, M. A. Armen, and H. Mabuchi, “Femtojoule-scale all-optical latching and modulation via cavity nonlinear optics,” Phys. Rev. Lett. 111, 203002 (2013).
[Crossref] [PubMed]

Baehr-Jones, T.

M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nature Photonics 4, 492–494 (2010).
[Crossref]

Bajcsy, M.

A. Majumdar, M. Bajcsy, D. Englund, and J. Vuckovic, “All optical switching with a single quantum dot strongly coupled to a photonic crystal cavity,” Selected Topics in Quantum Electronics, IEEE Journal of 18, 1812–1817 (2012).
[Crossref]

Barboza, A. P. M.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

Biermann, K.

Boyd, R.

R. Boyd, Nonlinear optics (Academic Press, 2008).

Buckley, S.

Caulfield, H. J.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nature Photonics 4, 261–263 (2014).
[Crossref]

Crespi, V.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

de Paula, A. M.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

Dolev, S.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nature Photonics 4, 261–263 (2014).
[Crossref]

Dong, J.

X. Wan, J. Dong, M. Qian, and W. Zhang, “Nonlinear optical properties of perovskite ymno3 studied by real-space recursion method,” Phys. Rev. B 61, 10664–10669 (2000).
[Crossref]

Eftekhar, A. A.

Elias, A. L.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Englund, D.

A. Majumdar, M. Bajcsy, D. Englund, and J. Vuckovic, “All optical switching with a single quantum dot strongly coupled to a photonic crystal cavity,” Selected Topics in Quantum Electronics, IEEE Journal of 18, 1812–1817 (2012).
[Crossref]

Fink, Y.

Gaeta, A. L.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7, 597607 (2013).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2005).

Gerace, D.

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Physical Review B 87235319 (2013).
[Crossref]

Gibbs, H.

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, 1985).

Hatami, F.

Hochberg, M.

M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nature Photonics 4, 492–494 (2010).
[Crossref]

Ibanescu, M.

Janisch, C.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Joannopoulos, J.

Johnson, S. G.

Kawaguchi, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Kwon, Y.-D.

Y.-D. Kwon, M. A. Armen, and H. Mabuchi, “Femtojoule-scale all-optical latching and modulation via cavity nonlinear optics,” Phys. Rev. Lett. 111, 203002 (2013).
[Crossref] [PubMed]

Lidorikis, E.

Lin, Z.

Lipson, M.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7, 597607 (2013).
[Crossref]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004).
[Crossref] [PubMed]

Liu, Z.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Ma, D.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Mabuchi, H.

Y.-D. Kwon, M. A. Armen, and H. Mabuchi, “Femtojoule-scale all-optical latching and modulation via cavity nonlinear optics,” Phys. Rev. Lett. 111, 203002 (2013).
[Crossref] [PubMed]

J. Wang, H. Mabuchi, and X.-L. Qi, “Calculation of divergent photon absorption in ultrathin films of a topological insulator,” Phys. Rev. B 88, 195127 (2013).
[Crossref]

Majumdar, A.

A. Majumdar and A. Rundquist, “Cavity-enabled self-electro-optic bistability in silicon photonics,” Opt. Lett. 39, 3864–3867 (2014).
[Crossref] [PubMed]

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Physical Review B 87235319 (2013).
[Crossref]

A. Majumdar, M. Bajcsy, D. Englund, and J. Vuckovic, “All optical switching with a single quantum dot strongly coupled to a photonic crystal cavity,” Selected Topics in Quantum Electronics, IEEE Journal of 18, 1812–1817 (2012).
[Crossref]

Mak, K. F.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

Malard, L. M.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

Masselink, W. T.

Matsuo, S.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Mehta, N.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Miller, D. A. B.

D. A. B. Miller, “Are optical transistors the logical next step?” Nature Photonics 4, 3–5 (2010).
[Crossref]

Miri, M.

Morandotti, R.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7, 597607 (2013).
[Crossref]

Moss, D. J.

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7, 597607 (2013).
[Crossref]

Notomi, M.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Nozaki, K.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Perea-Lopez, N.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Petykiewicz, J.

Qi, X.-L.

J. Wang, H. Mabuchi, and X.-L. Qi, “Calculation of divergent photon absorption in ultrathin films of a topological insulator,” Phys. Rev. B 88, 195127 (2013).
[Crossref]

Qian, M.

X. Wan, J. Dong, M. Qian, and W. Zhang, “Nonlinear optical properties of perovskite ymno3 studied by real-space recursion method,” Phys. Rev. B 61, 10664–10669 (2000).
[Crossref]

Radulaski, M.

Rivoire, K.

Rundquist, A.

Sato, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Sawchuk, A.

A. Sawchuk and T. C. Strand, “Digital optical computing,” Proceedings of the IEEE 72, 758–779 (1984).
[Crossref]

Segawa, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Shinya, A.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Sodagar, M.

Soljacic, M.

Strand, T. C.

A. Sawchuk and T. C. Strand, “Digital optical computing,” Proceedings of the IEEE 72, 758–779 (1984).
[Crossref]

Suzaki, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Takahashi, R.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

Terrones, M.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Vuckovic, J.

Wan, X.

X. Wan, J. Dong, M. Qian, and W. Zhang, “Nonlinear optical properties of perovskite ymno3 studied by real-space recursion method,” Phys. Rev. B 61, 10664–10669 (2000).
[Crossref]

Wang, J.

J. Wang, H. Mabuchi, and X.-L. Qi, “Calculation of divergent photon absorption in ultrathin films of a topological insulator,” Phys. Rev. B 88, 195127 (2013).
[Crossref]

Wang, Y.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Zhang, J. L.

Zhang, W.

X. Wan, J. Dong, M. Qian, and W. Zhang, “Nonlinear optical properties of perovskite ymno3 studied by real-space recursion method,” Phys. Rev. B 61, 10664–10669 (2000).
[Crossref]

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2005).

Nature Photonics (5)

D. A. B. Miller, “Are optical transistors the logical next step?” Nature Photonics 4, 3–5 (2010).
[Crossref]

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nature Photonics 4, 261–263 (2014).
[Crossref]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical ram based on nanocavities,” Nature Photonics 6, 248252 (2012).
[Crossref]

M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nature Photonics 4, 492–494 (2010).
[Crossref]

D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New cmos-compatible platforms based on silicon nitride and hydex for nonlinear optics,” Nature Photonics 7, 597607 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (3)

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from mos2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

J. Wang, H. Mabuchi, and X.-L. Qi, “Calculation of divergent photon absorption in ultrathin films of a topological insulator,” Phys. Rev. B 88, 195127 (2013).
[Crossref]

X. Wan, J. Dong, M. Qian, and W. Zhang, “Nonlinear optical properties of perovskite ymno3 studied by real-space recursion method,” Phys. Rev. B 61, 10664–10669 (2000).
[Crossref]

Phys. Rev. Lett. (1)

Y.-D. Kwon, M. A. Armen, and H. Mabuchi, “Femtojoule-scale all-optical latching and modulation via cavity nonlinear optics,” Phys. Rev. Lett. 111, 203002 (2013).
[Crossref] [PubMed]

Physical Review B (1)

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Physical Review B 87235319 (2013).
[Crossref]

Proceedings of the IEEE (1)

A. Sawchuk and T. C. Strand, “Digital optical computing,” Proceedings of the IEEE 72, 758–779 (1984).
[Crossref]

Scientific Reports (1)

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Scientific Reports 45530 (2010).

Selected Topics in Quantum Electronics, IEEE Journal of (1)

A. Majumdar, M. Bajcsy, D. Englund, and J. Vuckovic, “All optical switching with a single quantum dot strongly coupled to a photonic crystal cavity,” Selected Topics in Quantum Electronics, IEEE Journal of 18, 1812–1817 (2012).
[Crossref]

Other (3)

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2005).

R. Boyd, Nonlinear optics (Academic Press, 2008).

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, 1985).

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Figures (3)

Fig. 1
Fig. 1 (a) Schematic of a cavity with two modes with nonlinear interaction gnl. The losses in the cavity primarily arise from the leak rates in the reflection (κra) and transmission (κta) ports. There can be additional absorptive loss, which is neglected in our analysis. (b) At steady state, the output optical power is a bistable function of the input optical power. In practice, however, one observes a hysteretic behavior, depending on whether the input power is increased or decreased. The parameters for the simulation are: gnl/2π = 20 GHz; κta/2π = κra/2π = 3 GHz and Δa/κa = 8. (c) and (d) show different geometry of the nonlinear cavity: the cavity can itself be made of nonlinear material (c), or the cavity can be linear unto second-order, i.e., no χ(2) nonlinearity, but with a nonlinear material (e.g. 2D material or topological insulator) transferred on top of the cavity (d).
Fig. 2
Fig. 2 (a) The steady state bistability plot is used to identify the bias-points, around which one can modulate the input power to observe the change in the output power. (b) The ratio between the output power amplitude and input power amplitude as a function of Pbias and Pamp. (c) The frequency response for two different Pamp, showing the bandwidth changes depending on the amplitude. (d) Gain, defined as the switching ratio at a low frequency, (e) Bandwidth, defined at the 3 dB point, is plotted as a function of the Pamp. The parameters for the simulation are: gnl/2π = 20 GHz; κta/2π = κra/2π = 3 GHz and ∆a/κa = 8.
Fig. 3
Fig. 3 (a) The bias point Pbias as a function of the total linewidth 2κa and nonlinear interaction strength gnl. (b) log10(N), N being the intra-cavity photon number plotted as a function of κa and gnl. (c) Gain as a function of κa for different gnl. (d) Bandwidth as a function of κ for different gnl.

Equations (31)

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H ^ s = ω a a a + ω b b b + g n l [ b ( a ) 2 + b a 2 ] .
g n l = D ε 0 ( ω a 2 ε 0 ) ω b 2 ε 0 d r χ ( 2 ) ( r ) [ ε ( r ) ] 3 / 2 α a 2 ( r ) α b ( r ) .
g n l = ε 0 ( ω a ε 0 ε r ) 3 / 2 χ ( 2 ) V m
H r o t = Δ a a a + Δ b b b + g n l [ b ( a ) 2 + b a 2 ] + 2 κ r a E ( a + a ) .
d ρ d t = i [ H r o t , ρ ] + i = a , b κ i [ 2 A i ρ A i A i A i ρ ρ A i A i ] .
d a d t = i Δ a a ( κ r a + κ t a + κ l a ) a 2 i g n l b a + i 2 κ r a E ,
d b d t = i Δ b b ( κ r b + κ t b + κ l b ) b i g n l a 2 .
η 2 P t r a n s 3 + 2 η ( κ a 2 Δ a 2 ) P t r a n s 2 + ( Δ a 2 + κ a 2 ) 2 P t r a n s = 4 κ t a κ r a ( Δ a 2 + κ a 2 ) P i n ,
η 2 P t r a n s 3 + 2 η ( κ a 2 Δ a 2 ) P t r a n s 2 + ( Δ a 2 + κ a 2 ) 2 P t r a n s = κ 2 a ( Δ a 2 + κ a 2 ) P i n .
d a d t = i Δ a a ( κ r a + κ t a + κ l a ) a 2 i g n l b a + i 2 κ r a E , d b d t = i Δ b b ( κ r b + κ t b + κ l b ) b i g n l a 2 .
b = i g n l a 2 i Δ b ( κ r b + κ t b + κ l b )
i Δ a a ( κ r a + κ t a + κ l a ) a + 2 g n l 2 i Δ b ( κ r b + κ t b + κ l b ) a a 2 + i 2 κ r a E = 0
i Δ a a κ a a + 2 g n l 2 i Δ b κ b P t r a n s 2 κ t a a + i 2 κ r a E = 0
η 2 P t r a n s 3 + 2 η ( κ a 2 Δ a 2 ) P t r a n s 2 + ( Δ a 2 + κ a 2 ) P t r a n s = 4 κ t a κ r a ( Δ a 2 + κ a 2 ) P i n ,
d P i n d P t r a n s = 3 η 2 P t r a n s 2 + 4 η ( κ a 2 Δ a 2 ) P t r a n s + ( κ a 2 + Δ a 2 ) 2 = 0
P t r a n s c r = 2 ( Δ a 2 κ a 2 ) ± κ a 4 + Δ a 4 14 κ a 2 Δ a 2 3 η
η 2 P t r a n s 3 2 η Δ a 2 P t r a n s 2 + Δ a 4 P t r a n s = κ a 2 Δ a 2 P i n
3 η 2 P t r a n s 2 4 η Δ a 2 P t r a n s + Δ a 4 = 0
= Δ a a + χ a a a + 2 κ r E ( a + a )
d a d t = T r [ a d ρ d t ] = i Δ a ( κ r + κ t + κ l ) a + 2 i χ a a a + i 2 κ r E
i Δ a ( κ r + κ t + κ l ) a + 2 i χ a a a + i 2 κ r E = 0
a = i 2 κ r E i ( Δ + χ P t r a n s κ t ) ( κ r + κ t + κ l )
P t r a n s = 4 κ t κ r | E | 2 ( Δ + χ P t r a n s κ t ) 2 + ( κ r + κ t + κ l ) 2
η 2 P t r a n s 3 + 2 Δ η P t r a n s 2 + P t r a n s ( Δ 2 + ( κ r + κ t + κ l ) 2 ) 4 κ r κ t P i n = 0
d P i n d P t r a n s = 3 η 2 P t r a n s 2 + 4 Δ η P t r a n s + Δ 2 + ( κ r + κ t + κ l ) 2 = 0
P t r a n s c r = 2 Δ ± Δ 2 3 κ 2 3 η
P t r a n s c r = 2 Δ ( Δ 2 + 9 κ 2 ) 2 ( Δ 2 3 κ 2 ) 3 108 η κ t κ r
4 η 2 P t r a n s 3 + 4 η ( κ a κ b Δ a Δ b ) P t r a n s 2 + ( κ a 2 + Δ a 2 ) ( κ b 2 + Δ b 2 ) P t r a n s = 4 κ t a κ r a ( Δ b 2 + κ b 2 ) P i n
P t r a n s c r = κ a κ b 6 η [ ( Δ a κ a Δ b κ b 1 ) ± ( 1 Δ a κ a Δ b κ b ) 2 3 ( Δ a κ a + Δ b κ b ) 2 ]
| Δ a κ a Δ b κ b 1 | 3 | Δ a κ a + Δ b κ b |
Δ a κ a Δ b κ b 2 + 3

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