Abstract

We treat a PT-symmetric directional coupler with Kerr nonlinearity in both the single-mode waveguides. We investigate the quantum propagation of a coherent state through this device and simplify the appropriate description to a semiclassical one. Through numerical calculation we find that, for lower gain rates, the oscillatory behavior of the coupler changes only a little under the influence of the supplied noise. But for higher gain rates, the noise causes the coupler to behave like above the threshold of the increase of the photon number in one mode. In the coupler without the cross-Kerr interaction, this behavior is more pronounced. In this connection, the first- and second-order correlation functions exhibit a temporary oscillatory behavior for higher gain rates.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  45. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
    [Crossref]
  46. B. He, S.-B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
    [Crossref]
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  55. G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970).
    [Crossref]
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    [Crossref]
  57. G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. III. A generalized Wick theorem and multitime mapping,” Phys. Rev. D 2, 2206–2225 (1970).
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2018 (2)

S. Scheel and A. Szameit, “PT-symmetric photonic quantum systems with gain and loss do not exist,” Europhys. Lett. 122, 34001 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

2017 (4)

J. D. Huerta Morales and B. M. Rodríguez-Lara, “Photon propagation through linearly active dimers,” Appl. Sci. 7, 587 (2017).
[Crossref]

A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, and D. N. Christodoulides, “Dynamically encircling exceptional points: exact evolution and polarization state conversion,” Phys. Rev. Lett. 118, 093002 (2017).
[Crossref]

L. Zhang, G. S. Agarwal, W. P. Schleich, and M. O. Scully, “Hidden PT symmetry and quantization of a coupled-oscillator model of quantum amplification by superradiant emission of radiation,” Phys. Rev. A 96, 013827 (2017).
[Crossref]

P. Aleahmad, M. Khajavikhan, D. N. Christodoulides, and P. LiKamWa, “Integrated multi-port circulators for unidirectional optical information transport,” Sci. Rep. 7, 2129 (2017).
[Crossref]

2016 (4)

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

L. Ge and R. El-Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
[Crossref]

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

J. D. Huerta Morales, J. Guerrero, S. López-Aguayo, and B. M. Rodríguez-Lara, “Revisiting the optical PT-symmetric dimer,” Symmetry 8, 83 (2016).
[Crossref]

2015 (8)

B. He, S.-B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

T. J. Milburn, J. Doppler, C. A. Holmes, S. Portolan, S. Rotter, and P. Rabl, “General description of quasiadiabatic dynamical phenomena near exceptional points,” Phys. Rev. A 92, 052124 (2015).
[Crossref]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref]

T. Wasak, P. Szańkowski, V. V. Konotop, and M. Trippenbach, “Four-wave mixing in a parity-time (PT)-symmetric coupler,” Opt. Lett. 40, 5291–5294 (2015).
[Crossref]

R. El-Ganainy, J. I. Dadap, and R. M. Osgood, “Optical parametric amplification via non-Hermitian phase matching,” Opt. Lett. 40, 5086–5089 (2015).
[Crossref]

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of Bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref]

2014 (5)

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).
[Crossref]

M. Kreibich, J. Main, H. Cartarius, and G. Wunner, “Realizing PT-symmetric non-Hermiticity with ultracold atoms and Hermitian multiwell potentials,” Phys. Rev. A 90, 033630 (2014).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
[Crossref]

2013 (2)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

2012 (5)

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[Crossref]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

2011 (5)

G. Yoo, H.-S. Sim, and H. Schomerus, “Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators,” Phys. Rev. A 84, 063833 (2011).
[Crossref]

R. Uzdin, A. Mailybaev, and N. Moiseev, “On the observability and asymmetry of adiabatic state flips generated by exceptional points,” J. Phys. A Math. Theor. 44, 435302 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A Math. Theor. 44, 485302 (2011).
[Crossref]

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).
[Crossref]

2010 (5)

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

H. Schomerus, “Quantum noise and self-sustained radiation of PT-symmetric systems,” Phys. Rev. Lett. 104, 233601 (2010).
[Crossref]

2009 (2)

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglu, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

2008 (1)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 30402 (2008).
[Crossref]

2007 (2)

2005 (3)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46, 277–292 (2005).
[Crossref]

A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, “Geometric phase around exceptional points,” Phys. Rev. A 72, 014104 (2005).
[Crossref]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1992 (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

1983 (1)

F. Soto-Eguibar and P. Claverie, “Time evolution of the Wigner function,” J. Math. Phys. 24, 1104–1109 (1983).
[Crossref]

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

1970 (3)

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970).
[Crossref]

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. II. Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).
[Crossref]

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. III. A generalized Wick theorem and multitime mapping,” Phys. Rev. D 2, 2206–2225 (1970).
[Crossref]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Agarwal, G. S.

L. Zhang, G. S. Agarwal, W. P. Schleich, and M. O. Scully, “Hidden PT symmetry and quantization of a coupled-oscillator model of quantum amplification by superradiant emission of radiation,” Phys. Rev. A 96, 013827 (2017).
[Crossref]

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970).
[Crossref]

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[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. First-order correlation function versus time (in the units of ε 1 ) for the initial coherent state | ξ 1 , ξ 2 , where ξ 1 = 2 , ξ 2 = 0 , the nonlinear coupling parameter β = 0.01 ε , and (a)  γ = 0 , (b)  γ = 0.1 ε , (c)  γ = 0.6 ε , and (d)  γ = 0.7 ε .
Fig. 2.
Fig. 2. Second-order correlation function versus time (in the units of ε 1 ) for the initial coherent state | ξ 1 , ξ 2 , where ξ 1 = 2 , ξ 2 = 0 , the nonlinear coupling parameter β = 0.01 ε , and (a)  γ = 0 , (b)  γ = 0.1 ε , (c)  γ = 0.6 ε , and (d)  γ = 0.7 ε .
Fig. 3.
Fig. 3. First-order correlation function versus time (in the units of ε 1 ) for the initial coherent state | ξ 1 , ξ 2 , where ξ 1 = 2 , ξ 2 = 0 , without the nonlinear coupling, and for (a)  γ = 0 , (b)  γ = 0.1 ε , (c)  γ = 0.6 ε , and (d)  γ = 0.7 ε .
Fig. 4.
Fig. 4. Second-order correlation function versus time (in the units of ε 1 ) for the initial coherent state | ξ 1 , ξ 2 , where ξ 1 = 2 , ξ 2 = 0 , without the nonlinear coupling, and for (a)  γ = 0 , (b)  γ = 0.1 ε , (c)  γ = 0.6 ε , and (d)  γ = 0.7 ε .

Equations (20)

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d d t α 1 = ( i ω γ ) α 1 i ε α 2 i 2 β | α 1 | 2 α 1 i β 12 | α 2 | 2 α 1 , d d t α 2 = ( i ω + γ ) α 2 i ε α 1 i 2 β | α 2 | 2 α 2 i β 12 | α 1 | 2 α 2 ,
d d t ( α 1 α 2 ) = i H ( α 1 α 2 ) ,
H = ( ω i γ ε ε ω + i γ ) .
d d t a ^ 1 = i [ a ^ 1 , H ^ ] + f ^ 1 ( t ) , d d t a ^ 2 = i [ a ^ 2 , H ^ ] + f ^ 2 ( t ) ,
H ^ = ω a ^ 1 a ^ 1 + ω a ^ 2 a ^ 2 + β a ^ 1 2 a ^ 1 2 + β a ^ 2 2 a ^ 2 2 + β 12 a ^ 1 a ^ 1 a ^ 2 a ^ 2 + ε ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) ,
f ^ 1 ( t ) f ^ 1 ( t ) = 0 , f ^ 1 ( t ) f ^ 1 ( t ) = 2 γ δ ( t t ) ,
f ^ 2 ( t ) f ^ 2 ( t ) = 2 γ δ ( t t ) , f ^ 2 ( t ) f ^ 2 ( t ) = 0 .
d d t ρ ^ = i [ H ^ , ρ ^ ] + γ ( 2 a ^ 1 ρ ^ a ^ 1 a ^ 1 a ^ 1 ρ ^ ρ ^ a ^ 1 a ^ 1 ) + γ ( 2 a ^ 2 ρ ^ a ^ 2 a ^ 2 a ^ 2 ρ ^ ρ ^ a ^ 2 a ^ 2 ) ,
ρ ^ ( 0 ) = | ξ 1 , ξ 2 ξ 1 , ξ 2 | ,
ϕ S ( α 1 , α 2 ) = 1 π 4 Tr { ρ ^ exp [ j = 1 2 ( z j a ^ j z j * a ^ j ) ] } × exp [ j = 1 2 ( α j z j * α j * z j ) ] d 2 z 1 d 2 z 2 ,
t ϕ S = α 1 ( ( i H α 1 * + γ α 1 ) ϕ S ) + α 1 * ( ( i H α 1 + γ α 1 * ) ϕ S ) + α 2 ( ( i H α 2 * γ α 2 ) ϕ S ) + α 2 * ( ( i H α 2 γ α 2 * ) ϕ S ) + γ 2 α 1 * α 1 ϕ S + γ 2 α 2 * α 2 ϕ S + j = 1 2 i 1 2 β [ 3 α j * 2 α j ( α j * ϕ S ) 3 α j 2 α j * ( α j ϕ S ) ] + i 1 4 β 12 [ 3 α 1 α 1 * α 2 * ( α 2 * ϕ S ) + 3 α 1 * α 2 α 2 * ( α 1 * ϕ S ) 3 α 1 α 1 * α 2 ( α 2 ϕ S ) 3 α 1 α 2 α 2 * ( α 1 ϕ S ) ] ,
H ( α 1 , α 2 ) = ω | α 1 | 2 + ω | α 2 | 2 + β ( | α 1 | 4 2 | α 1 | 2 + 1 2 ) + β ( | α 2 | 4 2 | α 2 | 2 + 1 2 ) + β 12 ( | α 1 | 2 1 2 ) ( | α 2 | 2 1 2 ) + ε ( α 1 * α 2 + α 2 * α 1 ) .
ϕ S ( α 1 , α 2 , 0 ) = ( 2 π ) 2 exp ( 2 j = 1 2 | α j ξ j | 2 ) .
d d t α _ 1 = ( i ω γ ) α _ 1 i ε α _ 2 i 2 β ( | α _ 1 | 2 1 ) α _ 1 i β 12 ( | α _ 2 | 2 1 2 ) α _ 1 + L _ 1 ( t ) , d d t α _ 2 = ( i ω + γ ) α _ 2 i ε α _ 1 i 2 β ( | α _ 2 | 2 1 ) α _ 2 i β 12 ( | α _ 1 | 2 1 2 ) α _ 2 + L _ 2 ( t ) ,
L _ j ( t ) L _ j ( t ) = 0 ,
L _ j ( t ) L _ j * ( t ) = γ δ j j δ ( t t ) ,
M ^ = M ( α 1 , α 2 ) S = M ( α 1 , α 2 ) ϕ S ( α 1 , α 2 ) d 2 α 1 d 2 α 2
M ^ M ( α 1 , α 2 ) ϕ ¯ S ( α 1 , α 2 ) d 2 α 1 d 2 α 2
g 12 ( 1 ) = | a ^ 1 a ^ 2 | a ^ 1 a ^ 1 1 2 a ^ 2 a ^ 2 1 2 ,
g 12 ( 2 ) = a ^ 1 a ^ 2 a ^ 1 a ^ 2 a ^ 1 a ^ 1 a ^ 2 a ^ 2 .

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