Abstract

We study theoretically a parity-time (PT)-symmetric, saturable, balanced gain–loss system in a ring-cavity configuration. The saturable gain and loss are modeled by a two-level medium with or without population inversion. We show that the specifics of the spectral singularity can be fully controlled by the cavity and the atomic detuning parameters. The theory is based on the mean-field approximation, as in the standard theory of optical bistability. Further, in the linear regime we demonstrate the regularization of the singularity in detuned systems, while larger input power levels are shown to be adequate to limit the infinite growth in absence of detunings.

© 2014 Optical Society of America

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  1. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
  2. A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
    [CrossRef]
  3. A. Mostafazadeh, Phys. Rev. A 80, 032711 (2009).
    [CrossRef]
  4. Z. Ahmed, J. Phys. Math. Theory 42, 472005 (2009).
  5. S. Longhi, Phys. Rev. B 80, 165125 (2009).
  6. A. Mostafazadeh, Phys. Rev. A 83, 045801 (2011).
    [CrossRef]
  7. A. Mostafazadeh, Phys. Rev. Lett. 110, 260402 (2013).
    [CrossRef]
  8. X. Liu, S. Dutta Gupta, and G. S. Agarwal, Phys. Rev. A 89, 013824 (2014).
    [CrossRef]
  9. H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chéais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Léondel, Opt. Express 19, 18004 (2011).
    [CrossRef]
  10. A. Mostafazadeh, Phys. Rev. A 84, 023809 (2011).
    [CrossRef]
  11. A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 88, 033810 (2013).
    [CrossRef]
  12. A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 87, 063834 (2013).
    [CrossRef]
  13. A. Lupu, H. Benisty, and A. Degiron, Opt. Express 21, 21651 (2013).
    [CrossRef]
  14. B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
    [CrossRef]
  15. G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 30, 2764 (1984).
    [CrossRef]
  16. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
    [CrossRef]
  17. M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
    [CrossRef]
  18. S. Dutta Gupta, Nonlinear Optics of Stratified Media, Progress in Optics, E. Wolf, ed. (Elsevier Science, 1998), Vol. 38, pp. 1–84.
  19. Z. Ahmed, Phys. Rev. A 64, 042716 (2001).
    [CrossRef]
  20. G. S. Agarwal and S. Dutta Gupta, Opt. Lett. 27, 1205 (2002).
    [CrossRef]
  21. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
    [CrossRef]

2014 (2)

X. Liu, S. Dutta Gupta, and G. S. Agarwal, Phys. Rev. A 89, 013824 (2014).
[CrossRef]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

2013 (5)

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 88, 033810 (2013).
[CrossRef]

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 87, 063834 (2013).
[CrossRef]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

A. Mostafazadeh, Phys. Rev. Lett. 110, 260402 (2013).
[CrossRef]

A. Lupu, H. Benisty, and A. Degiron, Opt. Express 21, 21651 (2013).
[CrossRef]

2011 (4)

H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chéais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Léondel, Opt. Express 19, 18004 (2011).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 84, 023809 (2011).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 83, 045801 (2011).
[CrossRef]

2009 (4)

A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 80, 032711 (2009).
[CrossRef]

Z. Ahmed, J. Phys. Math. Theory 42, 472005 (2009).

S. Longhi, Phys. Rev. B 80, 165125 (2009).

2005 (1)

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

2002 (1)

2001 (1)

Z. Ahmed, Phys. Rev. A 64, 042716 (2001).
[CrossRef]

1986 (1)

M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
[CrossRef]

1984 (1)

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 30, 2764 (1984).
[CrossRef]

Agarwal, G. S.

X. Liu, S. Dutta Gupta, and G. S. Agarwal, Phys. Rev. A 89, 013824 (2014).
[CrossRef]

G. S. Agarwal and S. Dutta Gupta, Opt. Lett. 27, 1205 (2002).
[CrossRef]

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 30, 2764 (1984).
[CrossRef]

Ahmed, Z.

Z. Ahmed, J. Phys. Math. Theory 42, 472005 (2009).

Z. Ahmed, Phys. Rev. A 64, 042716 (2001).
[CrossRef]

Asquini, M. L.

M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
[CrossRef]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Barbillon, G.

Bender, C. M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Benisty, H.

Besbes, M.

Blaize, S.

Bruyant, A.

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Carmichael, H. J.

M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
[CrossRef]

Chéais, S.

Christodoulides, D. N.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

De Lustrac, A.

Degiron, A.

Delgado, F.

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

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Dutta Gupta, S.

X. Liu, S. Dutta Gupta, and G. S. Agarwal, Phys. Rev. A 89, 013824 (2014).
[CrossRef]

G. S. Agarwal and S. Dutta Gupta, Opt. Lett. 27, 1205 (2002).
[CrossRef]

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 30, 2764 (1984).
[CrossRef]

S. Dutta Gupta, Nonlinear Optics of Stratified Media, Progress in Optics, E. Wolf, ed. (Elsevier Science, 1998), Vol. 38, pp. 1–84.

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Fan, S.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Forget, S.

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Gianfreda, M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Joannopoulos, J. D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Lei, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Léondel, G.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Liu, X.

X. Liu, S. Dutta Gupta, and G. S. Agarwal, Phys. Rev. A 89, 013824 (2014).
[CrossRef]

Long, G. L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Longhi, S.

S. Longhi, Phys. Rev. B 80, 165125 (2009).

Lugiato, L. A.

M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
[CrossRef]

Lupu, A.

Melloni, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Monifi, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Mostafazadeh, A.

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 88, 033810 (2013).
[CrossRef]

A. Mostafazadeh, Phys. Rev. Lett. 110, 260402 (2013).
[CrossRef]

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 87, 063834 (2013).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 83, 045801 (2011).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 84, 023809 (2011).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 80, 032711 (2009).
[CrossRef]

A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[CrossRef]

Muga, J. G.

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

Narducci, L. M.

M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
[CrossRef]

Nori, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Özdemir, S. K.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Peng, B.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Popovic, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Ruschhaupt, A.

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

Sarisaman, M.

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 87, 063834 (2013).
[CrossRef]

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 88, 033810 (2013).
[CrossRef]

Vanwolleghem, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Yang, L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Yu, Z.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

J. Phys. A (1)

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

J. Phys. Math. Theory (1)

Z. Ahmed, J. Phys. Math. Theory 42, 472005 (2009).

Nat. Photonics (1)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, Nat. Photonics 7, 579 (2013).
[CrossRef]

Nat. Phys. (1)

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (9)

A. Mostafazadeh, Phys. Rev. A 80, 032711 (2009).
[CrossRef]

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 30, 2764 (1984).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 83, 045801 (2011).
[CrossRef]

M. L. Asquini, L. A. Lugiato, H. J. Carmichael, and L. M. Narducci, Phys. Rev. A 33, 360 (1986).
[CrossRef]

Z. Ahmed, Phys. Rev. A 64, 042716 (2001).
[CrossRef]

X. Liu, S. Dutta Gupta, and G. S. Agarwal, Phys. Rev. A 89, 013824 (2014).
[CrossRef]

A. Mostafazadeh, Phys. Rev. A 84, 023809 (2011).
[CrossRef]

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 88, 033810 (2013).
[CrossRef]

A. Mostafazadeh and M. Sarisaman, Phys. Rev. A 87, 063834 (2013).
[CrossRef]

Phys. Rev. B (1)

S. Longhi, Phys. Rev. B 80, 165125 (2009).

Phys. Rev. Lett. (3)

A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

A. Mostafazadeh, Phys. Rev. Lett. 110, 260402 (2013).
[CrossRef]

Other (1)

S. Dutta Gupta, Nonlinear Optics of Stratified Media, Progress in Optics, E. Wolf, ed. (Elsevier Science, 1998), Vol. 38, pp. 1–84.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic view of the PT-symmetric ring-cavity system with coupled cavities with balanced gain and loss medium.

Fig. 2.
Fig. 2.

Effect of the external cavity on the spectral singularity in absence of internal cavities (T=1.0): (a) δ=0.0, θ3=0.0, (b) δ=0.0, θ3=0.05, and (c) δ=0.001, θ3=0. Curves from right to left are for T3=1.0, 0.49, 0.09.

Fig. 3.
Fig. 3.

Effect of the internal cavities on the spectral singularity in absence of the external cavity (T3=1.0): (a) δ=0.0, θ=0.0, (b) δ=0.0, θ=0.05, and (c) δ=0.001, θ=0. Curves from right to left are for T=1.0, 0.5, 0.1.

Fig. 4.
Fig. 4.

Removal of the spectral singularity in detuned systems for T=T3=0.01 for (a) δ=0.0, 0.01, 0.05 with θ3=θ=0.0, (b) θ3=0.0, 0.05, 0.1 with θ=δ=0.0, and (c) θ=0.0, 0.0001, 0.0005 with θ3=δ=0.0.

Fig. 5.
Fig. 5.

Regularization of the spectral singularity by nonlinearity in a cavity configuration (T3=T=0.01, θ3=0.005 and δ=θ=0): log10(|x/y|) as a function of αL for different values of |y|, namely, |y|=106, 104, 103. Curves from right to left are for increasing values of |y|.

Fig. 6.
Fig. 6.

Regularization due to atomic detuning: log10(|x/y|) as a function of log10(|y|) (a) in a cavityless system (T3=T=1.0) with αL=0.75 for δ=0.0, 0.05, 0.2 and (b) in the presence of all the cavities T3=T=0.01 with θ3=θ=0.0 and αL=0.0002 for δ=0.0, 0.15, 0.5. Curves labeled 1–3 are for the increasing values of δ. In Fig. 6(b), the point marked by a cross on curve 2 corresponds to very high transmission at low input, and the thick solid curve shows the diode action corresponding to curve 2 in Fig. 6(b).

Fig. 7.
Fig. 7.

Effect of cavity detuning on spectral singularity in the presence of all the cavities T=T3=0.01 with δ=0.0, and αL=0.0002 for (a) θ=0, θ3=0.0, 0.2, 0.5 and (b) θ3=0, θ=0.0, 0.001, 0.003. Curves labeled by 1–3 are for increasing values of parameter of interest in each panel.

Equations (6)

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

y=x{TT3[τ2R3(1iT3θ3)]+ζτζ(1+δ2+|x|2)[τ+ζT3/T]1+δ2+|x[τ+ζT3/T]|2},
ζ=αL(1iδ)TT3(1+δ2+|x|2),τ=1+iRθ.
x=μ2EtTT3γγ,y=μ2Eiγγ,
δi=ωiω0,θi=(ωciω0)/κ,κ=cT/L.
y=x{T3Tα2L2TT3T},
αL=T3T2.

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