Abstract

We introduce a stochastic parity-time (PT)-symmetric coupler, which is based on dual-core waveguides with fluctuating parameters, such that the gain and the losses are exactly balanced in average. We consider different parametric regimes that correspond to the broken and unbroken PT symmetry, as well as to the exceptional point of the underlying deterministic system. We demonstrate that in all the cases the statistically averaged intensity of the field grows. This result holds for either linear or nonlinear coupler and is independent on the type of fluctuations.

© 2014 Optical Society of America

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  1. C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
    [CrossRef]
  2. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
    [CrossRef]
  3. W. D. Heiss, J. Phys. A 45, 444016 (2012).
    [CrossRef]
  4. A. Guo, G. J. Salamo, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
    [CrossRef]
  5. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
    [CrossRef]
  6. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  7. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
    [CrossRef]
  8. H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, Opt. Express 19, 18004 (2011).
    [CrossRef]
  9. C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
    [CrossRef]
  10. C. Hang, D. A. Zezyulin, V. V. Konotop, and G. Huang, Opt. Lett. 38, 4033 (2013).
    [CrossRef]
  11. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
    [CrossRef]
  12. C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
    [CrossRef]
  13. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
    [CrossRef]
  14. A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
    [CrossRef]
  15. L. Cruzeiro-Hansson, P. L. Christiansen, and J. N. Elgin, Phys. Rev. B 37, 7896 (1988).
    [CrossRef]
  16. V. V. Konotop and L. Vázquez, Nonlinear Random Waves (World Scientific, 1994).
  17. V. I. Klyatskin, Dynamics of Stochastic Systems (Elsevier, 2005).
  18. P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, J. Phys. A 46, 365201 (2013).
    [CrossRef]
  19. I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).
    [CrossRef]

2013

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
[CrossRef]

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, J. Phys. A 46, 365201 (2013).
[CrossRef]

I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).
[CrossRef]

C. Hang, D. A. Zezyulin, V. V. Konotop, and G. Huang, Opt. Lett. 38, 4033 (2013).
[CrossRef]

2012

W. D. Heiss, J. Phys. A 45, 444016 (2012).
[CrossRef]

2011

2010

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (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

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

2007

2005

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

1988

L. Cruzeiro-Hansson, P. L. Christiansen, and J. N. Elgin, Phys. Rev. B 37, 7896 (1988).
[CrossRef]

Aimez, V.

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

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).
[CrossRef]

Barbillon, G.

Bender, C. M.

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
[CrossRef]

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Bender, N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Benisty, H.

Berntson, B. K.

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
[CrossRef]

Besbes, M.

Blaize, S.

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Bruyant, A.

Chais, S.

Chen, Y.-F.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Christiansen, P. L.

L. Cruzeiro-Hansson, P. L. Christiansen, and J. N. Elgin, Phys. Rev. B 37, 7896 (1988).
[CrossRef]

Christodoulides, D. N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Cruzeiro-Hansson, L.

L. Cruzeiro-Hansson, P. L. Christiansen, and J. N. Elgin, Phys. Rev. B 37, 7896 (1988).
[CrossRef]

De Lustrac, A.

Degiron, A.

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

El-Ganainy, R.

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Elgin, J. N.

L. Cruzeiro-Hansson, P. L. Christiansen, and J. N. Elgin, Phys. Rev. B 37, 7896 (1988).
[CrossRef]

Ellis, F. M.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Factor, S.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Feng, L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Flach, S.

I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).
[CrossRef]

Forget, S.

Guo, A.

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

Hang, C.

C. Hang, D. A. Zezyulin, V. V. Konotop, and G. Huang, Opt. Lett. 38, 4033 (2013).
[CrossRef]

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

Heiss, W. D.

W. D. Heiss, J. Phys. A 45, 444016 (2012).
[CrossRef]

Huang, G.

C. Hang, D. A. Zezyulin, V. V. Konotop, and G. Huang, Opt. Lett. 38, 4033 (2013).
[CrossRef]

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Jackson, G. S.

I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).
[CrossRef]

Kevrekidis, P. G.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, J. Phys. A 46, 365201 (2013).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

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

Klyatskin, V. I.

V. I. Klyatskin, Dynamics of Stochastic Systems (Elsevier, 2005).

Konotop, V. V.

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

C. Hang, D. A. Zezyulin, V. V. Konotop, and G. Huang, Opt. Lett. 38, 4033 (2013).
[CrossRef]

V. V. Konotop and L. Vázquez, Nonlinear Random Waves (World Scientific, 1994).

Kottos, T.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

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

Lérondel, G.

Lu, M.-H.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Lupu, A.

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Musslimani, Z. H.

Parker, D.

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
[CrossRef]

Pelinovsky, D. E.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, J. Phys. A 46, 365201 (2013).
[CrossRef]

Ramezani, H.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

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

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Salamo, G. J.

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

Samuel, E.

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
[CrossRef]

Scherer, A.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Siviloglou, G. A.

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

Sukhorukov, A. A.

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

Tyugin, D. Y.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, J. Phys. A 46, 365201 (2013).
[CrossRef]

Vázquez, L.

V. V. Konotop and L. Vázquez, Nonlinear Random Waves (World Scientific, 1994).

Volatier-Ravat, M.

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

Xu, Y.-L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Xu, Z.

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

Zezyulin, D. A.

Am. J. Phys.

C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013).
[CrossRef]

J. Phys. A

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

W. D. Heiss, J. Phys. A 45, 444016 (2012).
[CrossRef]

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, J. Phys. A 46, 365201 (2013).
[CrossRef]

Nat. Phys.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

I. V. Barashenkov, G. S. Jackson, and S. Flach, Phys. Rev. A 88, 053817 (2013).
[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. B

L. Cruzeiro-Hansson, P. L. Christiansen, and J. N. Elgin, Phys. Rev. B 37, 7896 (1988).
[CrossRef]

Phys. Rev. Lett.

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

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Rep. Prog. Phys.

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Science

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, and A. Scherer, Science 333, 729 (2011).
[CrossRef]

Other

V. V. Konotop and L. Vázquez, Nonlinear Random Waves (World Scientific, 1994).

V. I. Klyatskin, Dynamics of Stochastic Systems (Elsevier, 2005).

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Figures (3)

Fig. 1.
Fig. 1.

(a) and (c) Evolution of the linear dimer (χ=0). (b) and (d) Evolution of the nonlinear dimer (χ=0.5). In panels (a) and (b) σΓ=0.15, σK=0, while in panels (c) and (d) σΓ=0, σK=0.15. In all panels γ=0.25, κ=1 and the initial conditions are S0(0)=S3(0)=1, S1(0)=S2(0)=0. The linear dynamics are obtained by means of analytical solution of system (13)–(15), while the nonlinear dynamics are obtained numerically, by averaging over 104 realizations. In panels (a) and (c) S1(z)0 and is not shown. The inset in (d) shows a semi-logarithmic plot of S0,3 together with the straight line that represents the dependence eμz with μ computed from Eq. (16).

Fig. 2.
Fig. 2.

Same as in Fig. 1 but with γ=1 (the exceptional point of the deterministic system). The insets in (b) and (d) show dependencies cos(Δφ) and sin(Δφ) (green and red curves, respectively) for a single arbitrarily chosen realization.

Fig. 3.
Fig. 3.

Same as in Fig. 2 but with γ=3 (the broken PT symmetry of the deterministic system).

Equations (21)

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

iq˙1=[κ+K(z)]q2+i[γ+Γ1(z)]q1+χ|q1|2q1,
iq˙2=[κ+K(z)]q1i[γ+Γ2(z)]q2+χ|q2|2q2.
Γ1,2(z)=K(z)=K(z)Γ1,2(z)=0,
Γj(z)Γk(z)=2δjkσΓ2δ(zz),
K(z)K(z)=2σK2δ(zz),
S0=|q1|2+|q2|2,S1=q1q2*+q1*q2,S3=|q1|2|q2|2,S2=i(q1q2*q1*q2),
S˙0=2γ(z)S0+2[γ+γ+(z)]S3,
S˙1=2γ(z)S1χS2S3,
S˙2=2γ(z)S22[κ+K(z)]S3+χS1S3,
S˙3=2γ(z)S3+2[γ+γ+(z)]S0+2[κ+K(z)]S2,
γ±(z)γ±(z)=σΓ2δ(zz),γ(z)γ+(z)=0.
x˙1=[γ+Γ1(z)]x1+[κ+K(z)]x2,x˙2=[γ+Γ2(z)]x2[κ+K(z)]x1.
K(z)xk(z)=0zK(z)K(z)δxk(z)δK(z)dz=(1)3kσK2x3k(z)(k=1,2)
(x˙1x˙2)=(σΓ2σK2+γκκσΓ2σK2γ)(x1x2).
Jlin=(q1q2*+q1*q2)exp(20zγ(z)dz),
S˙0=4σΓ2S0+2γS3,
S˙2=(2σΓ24σK2)S22κS3,
S˙3=2γS0+2κS2+4(σΓ2σK2)S3.
λ0=2σΓ2+2σΓ2κ2+2σK2γ2κ2γ2,λ±=±2iκ2γ2+3σΓ22σK2σΓ2γ2+2σK2κ2κ2γ2.
S0γCeμz,S3(σK4+γ2σK2)Ceμz,
μ=4σΓ22σK2+2σK4+γ2>0,

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