Abstract

The Zeno effect is investigated for soliton type pulses in a nonlinear directional coupler with dissipation. The effect consists in increase of the coupler transparency with increase of the dissipative losses in one of the arms. It is shown that localized dissipation can lead to switching of solitons between the arms. Power losses accompanying the switching can be fully compensated by using a combination of dissipative and active (in particular, parity-time-symmetric) segments.

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  1. M. Romagnioli, S. Trillo, and S. Wabnitz, Opt. Quantum Electron. 24, S1237 (1992).
    [CrossRef]
  2. Y. Chen, A. W. Snyder, and D. N. Pain, IEEE J. Quantum Electron. 28, 239 (1992).
    [CrossRef]
  3. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
    [CrossRef]
  4. A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
    [CrossRef]
  5. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
    [CrossRef] [PubMed]
  6. F. Kh. Abdullaev, V. V. Konotop, and V. S. Shchesnovich, Phys. Rev. A 83, 043811 (2011).
    [CrossRef]
  7. V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 81, 053611 (2010).
    [CrossRef]
  8. B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18, 756 (1977).
    [CrossRef]
  9. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 672 (1988).
    [CrossRef] [PubMed]
  10. C. Pare and M. Florjanczyk, Phys. Rev. A 41, 6287 (1990).
    [CrossRef] [PubMed]

2011 (1)

F. Kh. Abdullaev, V. V. Konotop, and V. S. Shchesnovich, Phys. Rev. A 83, 043811 (2011).
[CrossRef]

2010 (3)

V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 81, 053611 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

1992 (2)

M. Romagnioli, S. Trillo, and S. Wabnitz, Opt. Quantum Electron. 24, S1237 (1992).
[CrossRef]

Y. Chen, A. W. Snyder, and D. N. Pain, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

1990 (1)

C. Pare and M. Florjanczyk, Phys. Rev. A 41, 6287 (1990).
[CrossRef] [PubMed]

1988 (1)

1977 (1)

B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18, 756 (1977).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, V. V. Konotop, and V. S. Shchesnovich, Phys. Rev. A 83, 043811 (2011).
[CrossRef]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Chen, Y.

Y. Chen, A. W. Snyder, and D. N. Pain, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

Christodoulides, D. N.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Florjanczyk, M.

C. Pare and M. Florjanczyk, Phys. Rev. A 41, 6287 (1990).
[CrossRef] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Kivshar, Y. S.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Konotop, V. V.

F. Kh. Abdullaev, V. V. Konotop, and V. S. Shchesnovich, Phys. Rev. A 83, 043811 (2011).
[CrossRef]

V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 81, 053611 (2010).
[CrossRef]

Kottos, T.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Misra, B.

B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18, 756 (1977).
[CrossRef]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Pain, D. N.

Y. Chen, A. W. Snyder, and D. N. Pain, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

Pare, C.

C. Pare and M. Florjanczyk, Phys. Rev. A 41, 6287 (1990).
[CrossRef] [PubMed]

Ramezani, H.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Romagnioli, M.

M. Romagnioli, S. Trillo, and S. Wabnitz, Opt. Quantum Electron. 24, S1237 (1992).
[CrossRef]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Shchesnovich, V. S.

F. Kh. Abdullaev, V. V. Konotop, and V. S. Shchesnovich, Phys. Rev. A 83, 043811 (2011).
[CrossRef]

V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 81, 053611 (2010).
[CrossRef]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Snyder, A. W.

Y. Chen, A. W. Snyder, and D. N. Pain, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

Stegeman, G. I.

Sudarshan, E. C. G.

B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18, 756 (1977).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Trillo, S.

M. Romagnioli, S. Trillo, and S. Wabnitz, Opt. Quantum Electron. 24, S1237 (1992).
[CrossRef]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 672 (1988).
[CrossRef] [PubMed]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Wabnitz, S.

M. Romagnioli, S. Trillo, and S. Wabnitz, Opt. Quantum Electron. 24, S1237 (1992).
[CrossRef]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 672 (1988).
[CrossRef] [PubMed]

Wright, E. M.

Xu, Z.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Chen, A. W. Snyder, and D. N. Pain, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

J. Math. Phys. Sci. (1)

B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18, 756 (1977).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. Romagnioli, S. Trillo, and S. Wabnitz, Opt. Quantum Electron. 24, S1237 (1992).
[CrossRef]

Phys. Rev. A (5)

C. Pare and M. Florjanczyk, Phys. Rev. A 41, 6287 (1990).
[CrossRef] [PubMed]

F. Kh. Abdullaev, V. V. Konotop, and V. S. Shchesnovich, Phys. Rev. A 83, 043811 (2011).
[CrossRef]

V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 81, 053611 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Observation of a Zeno soliton in component u 1 for large enough damping ( γ 2 ) in component u 2 for the input pulse of the form u 1 ( z = 0 ) = sech ( τ / 2 ) , u 2 ( z = 0 ) = 0 . (a) Abrupt decay of the field u 1 for γ 2 = 1 . (b) A Zeno soliton in u 1 for γ 2 = 10 .

Fig. 2
Fig. 2

Distribution and decay of powers P j . (a) The exponential decay in the first arm and local energy transfer between the arms for the initial data as in Fig. 1 (i.e. below the threshold, A 1 2 < 6.7 ). Black solid curves show the full simulation of Eq. (1), while yellow (gray) dashed curves show the approximate results obtained from Eqs. (2a, 2b, 2c). (b) The same as in panel (a) but for the input amplitude A 1 = 5 , i.e. above the threshold, A 1 2 > 6.7 . The horizontal dashed line shows the normalized critical threshold value A cr 6.7 (i.e. the value 6.7 / 5 0.5 ). The sloop 2 / γ 2 is illustrated with the dashed-dotted line for γ 2 = 10 .

Fig. 3
Fig. 3

(a) The half-width at the half-maximum versus propagation distance for different input amplitudes A 1 ( a = 2 / A 1 ) ) and a fixed γ 2 = 10 . Solid (dashed) curves are for arm j = 1 (2). (b) The relative central phase ϕ ( z , τ = 0 ) obtained from (1) (solid curves) and from (2) (dashed curves), normalized with π.

Fig. 4
Fig. 4

Switching between the arms induced by localized defects. In (a) only a dissipative segment γ 1 ( z ) = Γ ( arctan [ 5 ( z z a ) ] arctan [ 5 ( z z b ) ] ) 2 with Γ = 0.135 , z a = 1.5 , and z b = 3.0 in the first arm is included. The inset shows the relative phase. The solid (blue) and dashed (red) curves show the simulations of the models in (1) and (2), respectively. In (b) the simultaneous effect of the dissipative and active segments, representing a PT -like structure, i.e. γ 2 ( z ) = γ 1 ( z ) , with γ 1 ( z ) as in panel (a) but with Γ = 0.065 . For the inset of (b) we have added a segment of gain/dissipation at a Δ z 10 distance and used engineered parameters in order to keep Q ( z ) 1 . In both (a) and (b) we use u 1 ( z = 0 ) = 20 sech ( 10 2 τ ) and u 2 ( z = 0 ) = 5 sech ( 5 τ / 2 ) , i.e. initially F ( 0 ) = 3 / 5 .

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

i u 1 z + u 1 τ τ + | u 1 | 2 u 1 + u 2 + i γ 1 u 1 = 0 , i u 2 z + u 2 τ τ + | u 2 | 2 u 2 + u 1 + i γ 2 u 2 = 0.
L = 2 A 1 2 a ϕ 1 , z 2 A 2 2 a ϕ 2 , z 2 a 3 ( A 1 4 + A 2 4 ) + 2 3 a ( A 1 2 + A 2 2 ) + 2 a A 1 A 2 cos ( ϕ 1 ϕ 2 ) ,
F z = γ ( 1 F 2 ) + 2 1 F 2 sin ( ϕ ) ,
ϕ z = δ F Q 2 F 1 F 2 cos ( ϕ ) ,
Q z = γ 1 Q ( 1 + F ) γ 2 Q ( 1 F ) .

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