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Quantum statistics of contradirectional Kerr nonlinear couplers
15 March 2000
Optics Communications 176 Ž2000. 149–154 www.elsevier.comrlocateroptcom
Quantum statistics of contradirectional Kerr nonlinear couplers G. Ariunbold 1, J. Perina ˇ
)
Department of Optics, Palacky´ UniÕersity, 17. listopadu 50, 772 07 Olomouc, Czech Republic Received 7 December 1999; accepted 13 January 2000
Abstract We study contradirectional couplers composed of two linearly coupled nonlinear Kerr waveguides. Intensity-dependent equations of motion are approximately solved, using a short-length coupler as an analogue of reference device. It is demonstrated that the switching is accompanied by quantum effects, such as two-mode squeezing of vacuum fluctuations and photon antibunching in single modes. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50; 42.60.Gd Keywords: Quantum statistics; Kerr effect; Contradirectional coupler; Switching
1. Introduction The codirectional and contradirectional propagation of light through two adjacent, parallel waveguides, has been greatly attracted in connection with the possibility of the switching, modulation and frequency selection of radiation in optics communications networks, over the last two decades w1x. The exchange of energy between two waveguides is possible because of the evanescent field overlap. The corresponding quantum generalization w2x of classical equations of motion and the convenience of the use of the space dependent momentum operator instead
) Corresponding author. Joint Laboratory of Optics of Palacky´ University and Physical Institute of Academy of Sciences of Czech Republic, 17. listopadu 50, 772 07 Olomouc, Czech Republic. E-mail: [email protected] 1 Permanent address: Department of Theoretical Physics, National University of Mongolia, 210646 Ulaanbaatar, Mongolia.
of time dependent Hamiltonian w3x enable us to investigate the nonclassical photon statistics w4x of linear w2,5x and nonlinear couplers w6x. An input–output formulation of the contradirectional coupler has been developed w3,7,8x. Using iterations in the interaction distance or a strong stimulating field in the second harmonic mode of the asymmetric nonlinear contradirectional coupler w9x, in general with the phase mismatch w10x, an effective transfer of nonclassical properties of light from nonlinear to linear waveguides was demonstrated and it was shown that the contrapropagating geometry provides better conditions for squeezing in single modes of the symmetric optical parametric nonlinear coupler w11x. The possibility of codirectional simulations of some contradirectional couplers was also examined w12x. For a review of quantum statistical properties of nonlinear couplers, see w13x. In the present paper, we study quantum statistics of the contradirectional coupler with two Kerr nonlinear waveguides. In Section 2 we derive system of
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 5 0 8 - 3
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
150
equations and their approximate solutions, Section 3 is devoted to the study of quantum statistical properties of the field modes, Section 4 represents a summary of the main conclusions and Appendix A contains details of the numerical method adopted in the text.
beam. Substituting new bosonic operators bˆ 1 z ,2 z s 1 Ž aˆ1 z " i aˆ 2 z ., we can rewrite Ž1. in the form
'2
d bˆ 1 z dz d bˆ 2 z dz
2. Description of the model The Kerr nonlinear coupler is composed of two parallel nonlinear waveguides with the third order nonlinear susceptibility, whose modes mutually exchange the energies by means of evanescent waves. In contrast to the codirectional Kerr coupler w14,15x in which both the modes are propagating towards, one can consider the contradirectional Kerr coupler with contrapropagating beams. The equations of motion of this Kerr coupler, in the interaction picture, are given by d aˆ1 z dz d aˆ 2 z dz
s i k aˆ 2 z q 2i g aˆ†1 zaˆ1 z aˆ1 z q i g˜ aˆ†2 zaˆ 2 z aˆ1 z ,
ž
/
ž
/
s yi k aˆ1 z y 2i g aˆ†2 zaˆ 2 z aˆ 2 z y i g˜ aˆ†1 zaˆ1 z aˆ 2 z ,
ž
/
ž
/
Ž 1.
s k bˆ 1 z q i gMˆ 1 zbˆ 1 z q fˆ2 z , s yk bˆ 2 z q i gMˆ 2 zbˆ 2 z q fˆ1 z ,
Ž 2.
where Mˆ 1 z ,2 z s Ž1 y g . bˆ †2 z ,1 zbˆ 1 z ,2 z q 2 bˆ †1 z ,2 zbˆ 2 z ,1 z and Nˆz s bˆ †1 zbˆ 2 z q bˆ †2 zbˆ 1 z s constant. The last term fˆ2 z s i g Ž1 q g . bˆ †2 zbˆ 22 z Ž fˆ1 z s i g Ž1 q g . bˆ †1 zbˆ 12 z . in the equation for the first Žsecond. mode defines a thirdorder nonlinear coupling of the second Žfirst. mode bˆ 2 z Ž bˆ 1 z ., representing an external source. Therefore, we assume that the single-mode solution of Eqs. Ž2. may be approximated by a solution of the secondorder nonlinear coupled equations, in which the higher-order source terms of the other mode are neglected. In this spirit, we see that bˆ †1 zbˆ 2 z , constant, bˆ †2 zbˆ 1 z , constant, and thus the two equations become decoupled in terms of bˆ 1 z ,2 z, but, of course, the equations for aˆ1 z ,2 z are coupled. The decoupled solutions are easily found for a given length L of the coupler in the form ˆ
bˆ 1 L s e " k Lqi g M 1 zL bˆ 1 0 , ˆ
bˆ 2 L s e " k Lqi g M 2 zL bˆ 2 0 ;
Ž 3.
where Nˆz s aˆ†1 zaˆ1 z y aˆ†2 zaˆ 2 z s constant. We denote the mode operators by aˆ1 z ,2 z for couplers with the length z and g Ž g˜ . being a self Žcross. nonlinear constant and k being linear coupling constant. To obtain Eqs. Ž1., we have used the following standard rules w13x for quantum contradirectional couplers: Ži. we write the corresponding momentum operator for codirectional case in the interaction picture for the Kerr coupler w15x
here z has been chosen such as z / 0, L and will be specified later. For contradirectional couplers, we assume that the first mode is specified on the lefthand side of one nonlinear waveguide, aˆ1 0 s aˆ1 , and the second mode is specified on the right-hand side of the other waveguide, aˆ 2 L s aˆ 2 , thus having
Gˆint s "gaˆ†1 2z aˆ12 z q "gaˆ†22z aˆ 22 z q "ga ˜ˆ†1 zaˆ1 zaˆ†2 zaˆ 2 z
ˆˆ1 , aˆ 2 0 s Cˆy1 aˆ 2 q iCˆy1 Sa
q Ž " k aˆ1 zaˆ†2 z q h.c. . , where the ratio g s gr2 ˜ g can be positive or negative depending upon the tensor nature of the thirdorder susceptibility w16x, Žii. we compose the Heisenberg equations for this process and finally, Žiii. we change the sign of the derivative for the backward
ˆ ˆ ˆy1 Sˆ. aˆ1 q i SC ˆ ˆy1 aˆ 2 , aˆ1 L s Cˆy1 Ž Cˆ 2 y CSC
ˆ
Ž 4. ˆ
where Cˆ s 12 Ž e k L e i g M 1 zL q ey k L e i g M 2 zL . and Sˆ ˆ ˆ s 12 Ž e k L e i g M 1 zL y ey k L e i g M 2 zL . with Mˆ 1 z ,2 z s Ž3 y ˆ ˆ ˆ g . J3 z . iŽ1 q g .Ž Jq z q Jy z .r2, where the angular momentum operators Jˆ3 z s 12 Ž aˆ†1 zaˆ1 z y aˆ†2 zaˆ 2 z ., Jˆq z s aˆ†1 zaˆ 2 z s Jˆy† z satisfy the commutation relations w Jˆ3 , Jˆ" x s "Jˆ" and w Jˆq , Jˆy x s 2 Jˆ3 . Provided z z z z z z that for contradirectional couplers Nˆz s 2 Jˆ3 z is the
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
constant of motion for couplers with any z, we can write Jˆ3 z ' Jˆ3 zX , so Jˆ1 zX commutes with all other ˆ ˆx s 0, ˆ operators with zX / z. Due to the fact that w C,S solutions Ž4. may be rewritten as
fulfilling n1Ž L. q n 2 Ž L. s < a 1 < 2 s constant. After a little algebra Žsee Appendix A., we can obtain `
n1 Ž L . , 4 < a 1 < 2
ˆ
=exp Ž y<2 k L Ž n q 1 .
Ž 5.
where uˆs g Ž3 y g . Jˆ3 zX ' g Ž3 y g . Jˆ3 z and fˆ s k q g Ž1 q g .Ž Jˆq z q Jˆy z .r2. It is clear that operators Ž5. satisfy the commutation relations w aˆ 2 0 ,1 L,aˆ†2 0 ,1 L x s 1ˆ and the conservation law aˆ†1 L aˆ1 L y aˆ†2 aˆ 2 s aˆ†2 0 aˆ 2 0 y aˆ†1 aˆ1. We have used four independent operators aˆ1,2 , aˆ1 z ,2 z which, of course, commute with each other.
This means that we have found the quantum consistent solution of the contradirectional coupler with the length L, using a reference coupler of the length z Ž z / 0, L.. Let z 0, then the solutions are simply found as aˆ1 z , aˆ1 , aˆ 2 z , aˆ 2 , since one can assume that no real interaction takes place in this sufficiently short coupler. It is worth to note that all above calculations will be accomplished exactly and simply in the special case when a self and cross nonlinearities compensate each other, where g s y1, which is analogue of the condition g s 1 for codirectional couplers w15x. In this case the forces fˆ1 z ,2 z are exactly neglected and, in addition, operator solutions Ž5. can be rewritten in terms of c-number hyperbolic functions, giving rise to the switching of two modes. This shows that the method adopted here is reasonable and, in general, we can expect nonclassical effects because of the operator form of hyperbolic functions involved.
™
y Ž sechh . Ž 1 y sechh . < a 1 < 2 < . ,
Ž 7.
where h s gL <1 q g <Ž n q 1.. In the same way, we can calculate two-mode quadrature Žusing the quadrature operators Xˆ1 s aˆ1 L q aˆ 2 0 q aˆ†1 L q aˆ†2 0 and Xˆ2 s Ž aˆ1 L q aˆ 2 0 y aˆ†1 L y aˆ†2 0 .ri. and principal squeeze variances w4x 2
² Ž D Xˆ1,2 . : s 2 1 q ² D aˆ†1 D aˆ1 : q ² D aˆ†2 D aˆ 2 : L L 0 0 2
q2Re² D aˆ†1 L D aˆ 2 0 : " Re ² Ž D aˆ1 L . :
ž
2
q² Ž D aˆ 2 0 . : q 2² D aˆ1 L D aˆ 2 0 :
/
,
lŽ L . s 2 1 q ² D aˆ†1 L D aˆ1 L : q ² D aˆ†2 0 D aˆ 2 0 : 2
q2Re² D aˆ†1 L D aˆ 2 0 : y <² Ž D aˆ1 L . : 2
q² Ž D aˆ 2 0 . : q 2² D aˆ1 L D aˆ 2 0 :< ,
Ž 8.
ˆ we denote ² D aˆ D bˆ : where, for two operators a, ˆ b, s ² ab ˆ ˆ : y ² aˆ: ² bˆ :. It is also worth to calculate the quadrature uncertainty product 2
2
u Ž L . s ² Ž D Xˆ1 . : ² Ž D Xˆ2 . : G 4 , testing generation of minimum uncertainty states. Fluctuations of the integrated intensity in single modes defined by the fourth-order moments in the field operators are given as w4x
3. Quantum effects in switching For the sake of simplicity, we investigate how the input vacuum in the second nonlinear waveguide is stimulated by the coherent state with an amplitude a 1 prepared in the first waveguide. The mean number of photons in single modes as a function of the length L is found to be
² Ž DW1
L ,2 0
.
2
: s ² aˆ†1 2 ,2 aˆ12 ,2 : y ² aˆ†1 ,2 :2² aˆ1 ,2 :2 L 0 L 0 L 0 L 0 s ²sech4 Ž fˆ L . : y ²sech2 Ž fˆ L . :2 .
Ž 9. Correlations of integrated intensity fluctuations in single modes are obtained as w4x
n1 Ž L . s ² aˆ†1 L aˆ1 L : s < a 1 < 2²sech2 Ž fˆ L . : , n 2 Ž L . s ² aˆ†2 0 aˆ 2 0 : s < a 1 < 2²tanh2 Ž fˆ L . : ,
n Ý Ž y1. Ž n q 1.
ns0
ˆ
aˆ1 L s e i u L sech Ž fˆ L . aˆ1 q itanh Ž fˆ L . aˆ 2 , aˆ 2 0 s eyi u L sech Ž fˆ L . aˆ 2 q itanh Ž fˆ L . aˆ1 ,
151
Ž 6.
2
² D aˆ†1 aˆ1 D aˆ†2 aˆ 2 : ' y² Ž DW1 ,2 . : , L L 0 0 L 0
Ž 10 .
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
152
verifying that the total variance is ² Ž DW1
L
.
2
: q ² Ž D W2
0
.
2
:
q 2² D aˆ†1 L aˆ1 L D aˆ†2 0 aˆ 2 0 :s0 ,
Ž 11 .
which indicates that the whole field is Poissonian Žwhile sub-Poissonian statistics in single modes can be observed.. Thus the nonlinear interaction modifies only the phase of the compound field leading to squeezing of vacuum fluctuations, whereas the photon statistics are Poissonian, as is typical for the Kerr effect w17x. We have depicted in Fig. 1 all the above statistical quantities in dependence on the length L of the coupler. We have chosen all constants of the contradirectional coupler as those used in w14x for the codirectional case, k s 1, g s 0.25, g s 1, input two-mode coherent state being < a 1 , a 2 :, a 1 s 2.5, a 2 s 0. One can see from Fig. 1 that the switching of modes is accompanied by an amplitude collapse around L s 1, when the initial zero fluctuations in single modes
increase to their maximum, representing photon bunching, whereas, the correlation of these single mode fluctuations decreases to its minimum ŽFig. 1e., showing the mode anticorrelation. At the same time, we see from Figs. 1b–d that a two-mode 2 squeezed state is generated, since ² Ž D Xˆ2 . : - 2, or lŽ L. - 2 and uŽ L. , 4 for L , 1. In the following stage, the correlation of single mode fluctuations increase, while quantum fluctuations in single modes reduce and the variances are negative ŽFig. 1e. showing photon antibunching and finally, they approach again zero value corresponding to the coherent state. As is pointed out in w14,15x, we can also determine an effective nonlinearity parameter as heff s g <1 q g <Ž< a 1 < 2 q < a 2 < 2 .rk , so that in the limit heff 0 the system becomes linear and there occurs only classical switching, the form of which is not affected by the nonlinear coupling, similarly as, for instance, when g s y1 or g < k for rather low intensity of the initial coherent state. The presence of strong nonlinearity heff ` Ži.e., k 0. leads to the selftrapping effect as a consequence of the occurrence of two decoupled Kerr nonlinear waveguides.
™
™
™
4. Conclusions
Fig. 1. Quantum statistical characteristics in the contradirectional Kerr coupler as functions of the length L with the coupling constants k s1, g s 0.25, g s1, for input two-mode coherent state < a 1 , a 2 :, a 1 s 2.5, a 2 s 0. Ža. Evolution of the mean number of photons n1Ž L. Žsolid curve. and n 2 Ž L. Ždotted curve.; Žb. 2 two-mode quadrature variances ² Ž D Xˆ1 . : Žsolid curve. and 2 ² Ž D Xˆ2 . : Ždotted curve.; Žc. two-mode principle squeeze variance lŽ L.; Žd. two-mode uncertainty product uŽ L.; Že. correlations ² D aˆ†1 Laˆ1 L D aˆ†2 0 aˆ 2 0 : Žsolid curve. of integrated intensity fluctuations and fluctuations ²Ž DW1 L,2 0 . 2 : Ždotted curve. of the integrated intensity.
We have investigated the quantum contradirectional coupler composed of two Kerr nonlinear waveguides. We have found approximate solutions of intensity-dependent operator equations. Using operator substitutions, we have obtained Eqs. Ž2. containing linear and nonlinear coupling terms and external higher-order nonlinear source terms. The single-mode solution of the exact equations may be approximated by a solution of weakly coupled equations, whereas the higher-order nonlinear coupling of the other mode can be neglected. This enables us to extract at least main quantum features of the contradirectional Kerr coupler. In fact, a similar idea has been adopted, for instance, in w15,18x as rotating wave approximation. In doing so, we argue that even for contradirectional couplers, this aspect does still work as well. Moreover, the formulation w13x that the operator solutions are written for the input and output of the coupler Žnot inside of the volume., automatically ensures the conservation of the boson commutation rules for all operators involved in Ž1..
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
Then, when we find a constant of motion of equations, it is constant for all couplers with different lengths. A sufficiently short coupler, approximate solutions of which are known, greatly simplifies the whole problem in such a way that its operators commute with the operators of couplers with different lengths, i.e., it plays a role as a ’reference’ device. Using the quantum consistent solutions, we have calculated the mean number of photons, fluctuations of the integrated intensity and their correlation in single modes and have demonstrated the quantum effect of collapse during switching of the modes, accompanied by enhanced squeezed quantum fluctuations. Two-mode quadrature squeezing and subPoissonian photon statistics are also obtained in this contradirectional device.
Acknowledgements
153
where
L "s
2 l " sinhh 2h coshh y l3 sinhh
ž
L3 s coshh y
, y2
l3 2h
sinhh
/
,
h 2 s 14 l23 q lq ly , we have `
n1 s 4 < a 1 < 2
n Ý Ž y1. Ž n q 1. ey2 k LŽ nq1.
ns0
ˆ ˆ ˆ =² a 1 ,0
here the coefficients read h s gL <1 q g <Ž n q 1., < L < s tanhh , L 3 s sech2h. Using the fact that Jˆ3 z s constant, its exponent may be put on the left-hand side changing its sub-index and returning again to z Žremember that aˆ1 ,2 , aˆ1,2 ., it holds that z z `
This work was supported by Complex Grant No VS96028 and Research Project CEZ: J14r98 of the Czech Ministry of Education.
n1 , 4 < a 1 < 2
n Ý Ž y1.
ns0
= Ž n q 1 . ey2 k LŽ nq1. eŽexpŽln'L 3 .y1.< a 1 < )
=² a 1 ,0
We numerically calculate the mean number of photons Ž6. expanding the function f Ž x . s sech2 r s 4rŽ x q 1rx q 2. with x s ey2 r Ž r G 0. into Taylor series as ` n Ý Ž y1. Ž n q 1.
ns0
ˆ =² a 1 ,0
Ž A.1 .
If we use the ordering theorem for the angular momentum operators w19x exp lq Jˆqq ly Jˆyq l3 Jˆ3
ž
†
e L aˆ 2 a 1 < a 1 ,0: .
Ž A.3 .
Using the Baker–Hausdorff formula for the second mode, we obtain the expression Ž7..
Appendix A
n1 s 4 < a 1 < 2
aˆ 2
2
/
s exp Ž Lq Jˆq . exp Ž ln L3 . Jˆ3 exp Ž Ly Jˆy . ,
Ž A.2 .
References w1x A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984. w2x J. Janszky, C. Sibilia, M. Bertolotti, J. Mod. Opt. 38 Ž1991. 2467. w3x M. Toren, Y. Ben-Aryeh, Quantum Opt. 6 Ž1994. 425. w4x J. Perina, ˇ Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd edn., Kluwer, Dordrecht, 1991. w5x W.K. Lai, V. Buzek, ˇ Knight, Phys. Rev. A 43 Ž1991. 6323. w6x G. Assanto, A. Laureti-Palma, C. Sibilia, M. Bertolotti, Opt. Commun. 110 Ž1994. 599. w7x J. Perina, J. Perina ˇ ˇ Jr., Quantum Semiclass. Opt. 7 Ž1995. 849. w8x A. Luis, Perina, Quantum Semiclass. Opt. 8 Ž1996. 39. ˇ w9x J. Perina, Acta Phys. Slov. 45 Ž1995. 279. ˇ w10x J. Perina, J. Perina ˇ ˇ Jr., Quantum Semiclass. Opt. 7 Ž1995. 863. w11x N. Korolkova, J. Perina, Opt. Commun. 137 Ž1997. 263. ˇ ˇ ˇ L. Mista w12x J. Rehacek, J. Mod. Opt. 46 Ž1999. ˇ Jr., J. Perina, ˇ 801.
154
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
w13x J. Perina Progress in Optics, Vol. 41, E. Wolf ˇ Jr., J. Perina, ˇ ŽEd.., Elsevier, Amsterdam, in print. w14x A. Chefles, S.M. Barnett, J. Mod. Opt. 43 Ž1996. 709; J. Fiurasek, J. Perina, Opt. Commun. 167 Ž1999. ´ˇ J. Krepelka, ˇ ˇ 115. w15x N. Korolkova, J. Perina, Opt. Commun. 136 Ž1996. 135; J. ˇ Mod. Opt. 44 Ž1997. 1525. w16x R.W. Boyd, Nonlinear Optics, Academic Press, Boston, 1992, chapter 4.
w17x R. Tanas, ´ Coherence and Quantum Optics 5, L. Mandel, E. Wolf ŽEds.., Plenum, New York, 1983, p. 645. w18x A. Mecozzi, S. Trillo, S. Wabnitz, B. Daino, Opt. Lett. 4 Ž1987. 275; A.P. Alodjants, S.M. Arakelian, A.S. Chirkin, Quantum Semiclass. Opt. 9 Ž1997. 311. w19x M. Ban, J. Math. Phys. 33 Ž1992. 3213; J. Opt. Soc. Am. B 10 Ž1993. 1347.
Optics Communications 176 Ž2000. 149–154 www.elsevier.comrlocateroptcom
Quantum statistics of contradirectional Kerr nonlinear couplers G. Ariunbold 1, J. Perina ˇ
)
Department of Optics, Palacky´ UniÕersity, 17. listopadu 50, 772 07 Olomouc, Czech Republic Received 7 December 1999; accepted 13 January 2000
Abstract We study contradirectional couplers composed of two linearly coupled nonlinear Kerr waveguides. Intensity-dependent equations of motion are approximately solved, using a short-length coupler as an analogue of reference device. It is demonstrated that the switching is accompanied by quantum effects, such as two-mode squeezing of vacuum fluctuations and photon antibunching in single modes. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50; 42.60.Gd Keywords: Quantum statistics; Kerr effect; Contradirectional coupler; Switching
1. Introduction The codirectional and contradirectional propagation of light through two adjacent, parallel waveguides, has been greatly attracted in connection with the possibility of the switching, modulation and frequency selection of radiation in optics communications networks, over the last two decades w1x. The exchange of energy between two waveguides is possible because of the evanescent field overlap. The corresponding quantum generalization w2x of classical equations of motion and the convenience of the use of the space dependent momentum operator instead
) Corresponding author. Joint Laboratory of Optics of Palacky´ University and Physical Institute of Academy of Sciences of Czech Republic, 17. listopadu 50, 772 07 Olomouc, Czech Republic. E-mail: [email protected] 1 Permanent address: Department of Theoretical Physics, National University of Mongolia, 210646 Ulaanbaatar, Mongolia.
of time dependent Hamiltonian w3x enable us to investigate the nonclassical photon statistics w4x of linear w2,5x and nonlinear couplers w6x. An input–output formulation of the contradirectional coupler has been developed w3,7,8x. Using iterations in the interaction distance or a strong stimulating field in the second harmonic mode of the asymmetric nonlinear contradirectional coupler w9x, in general with the phase mismatch w10x, an effective transfer of nonclassical properties of light from nonlinear to linear waveguides was demonstrated and it was shown that the contrapropagating geometry provides better conditions for squeezing in single modes of the symmetric optical parametric nonlinear coupler w11x. The possibility of codirectional simulations of some contradirectional couplers was also examined w12x. For a review of quantum statistical properties of nonlinear couplers, see w13x. In the present paper, we study quantum statistics of the contradirectional coupler with two Kerr nonlinear waveguides. In Section 2 we derive system of
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 5 0 8 - 3
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
150
equations and their approximate solutions, Section 3 is devoted to the study of quantum statistical properties of the field modes, Section 4 represents a summary of the main conclusions and Appendix A contains details of the numerical method adopted in the text.
beam. Substituting new bosonic operators bˆ 1 z ,2 z s 1 Ž aˆ1 z " i aˆ 2 z ., we can rewrite Ž1. in the form
'2
d bˆ 1 z dz d bˆ 2 z dz
2. Description of the model The Kerr nonlinear coupler is composed of two parallel nonlinear waveguides with the third order nonlinear susceptibility, whose modes mutually exchange the energies by means of evanescent waves. In contrast to the codirectional Kerr coupler w14,15x in which both the modes are propagating towards, one can consider the contradirectional Kerr coupler with contrapropagating beams. The equations of motion of this Kerr coupler, in the interaction picture, are given by d aˆ1 z dz d aˆ 2 z dz
s i k aˆ 2 z q 2i g aˆ†1 zaˆ1 z aˆ1 z q i g˜ aˆ†2 zaˆ 2 z aˆ1 z ,
ž
/
ž
/
s yi k aˆ1 z y 2i g aˆ†2 zaˆ 2 z aˆ 2 z y i g˜ aˆ†1 zaˆ1 z aˆ 2 z ,
ž
/
ž
/
Ž 1.
s k bˆ 1 z q i gMˆ 1 zbˆ 1 z q fˆ2 z , s yk bˆ 2 z q i gMˆ 2 zbˆ 2 z q fˆ1 z ,
Ž 2.
where Mˆ 1 z ,2 z s Ž1 y g . bˆ †2 z ,1 zbˆ 1 z ,2 z q 2 bˆ †1 z ,2 zbˆ 2 z ,1 z and Nˆz s bˆ †1 zbˆ 2 z q bˆ †2 zbˆ 1 z s constant. The last term fˆ2 z s i g Ž1 q g . bˆ †2 zbˆ 22 z Ž fˆ1 z s i g Ž1 q g . bˆ †1 zbˆ 12 z . in the equation for the first Žsecond. mode defines a thirdorder nonlinear coupling of the second Žfirst. mode bˆ 2 z Ž bˆ 1 z ., representing an external source. Therefore, we assume that the single-mode solution of Eqs. Ž2. may be approximated by a solution of the secondorder nonlinear coupled equations, in which the higher-order source terms of the other mode are neglected. In this spirit, we see that bˆ †1 zbˆ 2 z , constant, bˆ †2 zbˆ 1 z , constant, and thus the two equations become decoupled in terms of bˆ 1 z ,2 z, but, of course, the equations for aˆ1 z ,2 z are coupled. The decoupled solutions are easily found for a given length L of the coupler in the form ˆ
bˆ 1 L s e " k Lqi g M 1 zL bˆ 1 0 , ˆ
bˆ 2 L s e " k Lqi g M 2 zL bˆ 2 0 ;
Ž 3.
where Nˆz s aˆ†1 zaˆ1 z y aˆ†2 zaˆ 2 z s constant. We denote the mode operators by aˆ1 z ,2 z for couplers with the length z and g Ž g˜ . being a self Žcross. nonlinear constant and k being linear coupling constant. To obtain Eqs. Ž1., we have used the following standard rules w13x for quantum contradirectional couplers: Ži. we write the corresponding momentum operator for codirectional case in the interaction picture for the Kerr coupler w15x
here z has been chosen such as z / 0, L and will be specified later. For contradirectional couplers, we assume that the first mode is specified on the lefthand side of one nonlinear waveguide, aˆ1 0 s aˆ1 , and the second mode is specified on the right-hand side of the other waveguide, aˆ 2 L s aˆ 2 , thus having
Gˆint s "gaˆ†1 2z aˆ12 z q "gaˆ†22z aˆ 22 z q "ga ˜ˆ†1 zaˆ1 zaˆ†2 zaˆ 2 z
ˆˆ1 , aˆ 2 0 s Cˆy1 aˆ 2 q iCˆy1 Sa
q Ž " k aˆ1 zaˆ†2 z q h.c. . , where the ratio g s gr2 ˜ g can be positive or negative depending upon the tensor nature of the thirdorder susceptibility w16x, Žii. we compose the Heisenberg equations for this process and finally, Žiii. we change the sign of the derivative for the backward
ˆ ˆ ˆy1 Sˆ. aˆ1 q i SC ˆ ˆy1 aˆ 2 , aˆ1 L s Cˆy1 Ž Cˆ 2 y CSC
ˆ
Ž 4. ˆ
where Cˆ s 12 Ž e k L e i g M 1 zL q ey k L e i g M 2 zL . and Sˆ ˆ ˆ s 12 Ž e k L e i g M 1 zL y ey k L e i g M 2 zL . with Mˆ 1 z ,2 z s Ž3 y ˆ ˆ ˆ g . J3 z . iŽ1 q g .Ž Jq z q Jy z .r2, where the angular momentum operators Jˆ3 z s 12 Ž aˆ†1 zaˆ1 z y aˆ†2 zaˆ 2 z ., Jˆq z s aˆ†1 zaˆ 2 z s Jˆy† z satisfy the commutation relations w Jˆ3 , Jˆ" x s "Jˆ" and w Jˆq , Jˆy x s 2 Jˆ3 . Provided z z z z z z that for contradirectional couplers Nˆz s 2 Jˆ3 z is the
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
constant of motion for couplers with any z, we can write Jˆ3 z ' Jˆ3 zX , so Jˆ1 zX commutes with all other ˆ ˆx s 0, ˆ operators with zX / z. Due to the fact that w C,S solutions Ž4. may be rewritten as
fulfilling n1Ž L. q n 2 Ž L. s < a 1 < 2 s constant. After a little algebra Žsee Appendix A., we can obtain `
n1 Ž L . , 4 < a 1 < 2
ˆ
=exp Ž y<2 k L Ž n q 1 .
Ž 5.
where uˆs g Ž3 y g . Jˆ3 zX ' g Ž3 y g . Jˆ3 z and fˆ s k q g Ž1 q g .Ž Jˆq z q Jˆy z .r2. It is clear that operators Ž5. satisfy the commutation relations w aˆ 2 0 ,1 L,aˆ†2 0 ,1 L x s 1ˆ and the conservation law aˆ†1 L aˆ1 L y aˆ†2 aˆ 2 s aˆ†2 0 aˆ 2 0 y aˆ†1 aˆ1. We have used four independent operators aˆ1,2 , aˆ1 z ,2 z which, of course, commute with each other.
This means that we have found the quantum consistent solution of the contradirectional coupler with the length L, using a reference coupler of the length z Ž z / 0, L.. Let z 0, then the solutions are simply found as aˆ1 z , aˆ1 , aˆ 2 z , aˆ 2 , since one can assume that no real interaction takes place in this sufficiently short coupler. It is worth to note that all above calculations will be accomplished exactly and simply in the special case when a self and cross nonlinearities compensate each other, where g s y1, which is analogue of the condition g s 1 for codirectional couplers w15x. In this case the forces fˆ1 z ,2 z are exactly neglected and, in addition, operator solutions Ž5. can be rewritten in terms of c-number hyperbolic functions, giving rise to the switching of two modes. This shows that the method adopted here is reasonable and, in general, we can expect nonclassical effects because of the operator form of hyperbolic functions involved.
™
y Ž sechh . Ž 1 y sechh . < a 1 < 2 < . ,
Ž 7.
where h s gL <1 q g <Ž n q 1.. In the same way, we can calculate two-mode quadrature Žusing the quadrature operators Xˆ1 s aˆ1 L q aˆ 2 0 q aˆ†1 L q aˆ†2 0 and Xˆ2 s Ž aˆ1 L q aˆ 2 0 y aˆ†1 L y aˆ†2 0 .ri. and principal squeeze variances w4x 2
² Ž D Xˆ1,2 . : s 2 1 q ² D aˆ†1 D aˆ1 : q ² D aˆ†2 D aˆ 2 : L L 0 0 2
q2Re² D aˆ†1 L D aˆ 2 0 : " Re ² Ž D aˆ1 L . :
ž
2
q² Ž D aˆ 2 0 . : q 2² D aˆ1 L D aˆ 2 0 :
/
,
lŽ L . s 2 1 q ² D aˆ†1 L D aˆ1 L : q ² D aˆ†2 0 D aˆ 2 0 : 2
q2Re² D aˆ†1 L D aˆ 2 0 : y <² Ž D aˆ1 L . : 2
q² Ž D aˆ 2 0 . : q 2² D aˆ1 L D aˆ 2 0 :< ,
Ž 8.
ˆ we denote ² D aˆ D bˆ : where, for two operators a, ˆ b, s ² ab ˆ ˆ : y ² aˆ: ² bˆ :. It is also worth to calculate the quadrature uncertainty product 2
2
u Ž L . s ² Ž D Xˆ1 . : ² Ž D Xˆ2 . : G 4 , testing generation of minimum uncertainty states. Fluctuations of the integrated intensity in single modes defined by the fourth-order moments in the field operators are given as w4x
3. Quantum effects in switching For the sake of simplicity, we investigate how the input vacuum in the second nonlinear waveguide is stimulated by the coherent state with an amplitude a 1 prepared in the first waveguide. The mean number of photons in single modes as a function of the length L is found to be
² Ž DW1
L ,2 0
.
2
: s ² aˆ†1 2 ,2 aˆ12 ,2 : y ² aˆ†1 ,2 :2² aˆ1 ,2 :2 L 0 L 0 L 0 L 0 s ²sech4 Ž fˆ L . : y ²sech2 Ž fˆ L . :2 .
Ž 9. Correlations of integrated intensity fluctuations in single modes are obtained as w4x
n1 Ž L . s ² aˆ†1 L aˆ1 L : s < a 1 < 2²sech2 Ž fˆ L . : , n 2 Ž L . s ² aˆ†2 0 aˆ 2 0 : s < a 1 < 2²tanh2 Ž fˆ L . : ,
n Ý Ž y1. Ž n q 1.
ns0
ˆ
aˆ1 L s e i u L sech Ž fˆ L . aˆ1 q itanh Ž fˆ L . aˆ 2 , aˆ 2 0 s eyi u L sech Ž fˆ L . aˆ 2 q itanh Ž fˆ L . aˆ1 ,
151
Ž 6.
2
² D aˆ†1 aˆ1 D aˆ†2 aˆ 2 : ' y² Ž DW1 ,2 . : , L L 0 0 L 0
Ž 10 .
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
152
verifying that the total variance is ² Ž DW1
L
.
2
: q ² Ž D W2
0
.
2
:
q 2² D aˆ†1 L aˆ1 L D aˆ†2 0 aˆ 2 0 :s0 ,
Ž 11 .
which indicates that the whole field is Poissonian Žwhile sub-Poissonian statistics in single modes can be observed.. Thus the nonlinear interaction modifies only the phase of the compound field leading to squeezing of vacuum fluctuations, whereas the photon statistics are Poissonian, as is typical for the Kerr effect w17x. We have depicted in Fig. 1 all the above statistical quantities in dependence on the length L of the coupler. We have chosen all constants of the contradirectional coupler as those used in w14x for the codirectional case, k s 1, g s 0.25, g s 1, input two-mode coherent state being < a 1 , a 2 :, a 1 s 2.5, a 2 s 0. One can see from Fig. 1 that the switching of modes is accompanied by an amplitude collapse around L s 1, when the initial zero fluctuations in single modes
increase to their maximum, representing photon bunching, whereas, the correlation of these single mode fluctuations decreases to its minimum ŽFig. 1e., showing the mode anticorrelation. At the same time, we see from Figs. 1b–d that a two-mode 2 squeezed state is generated, since ² Ž D Xˆ2 . : - 2, or lŽ L. - 2 and uŽ L. , 4 for L , 1. In the following stage, the correlation of single mode fluctuations increase, while quantum fluctuations in single modes reduce and the variances are negative ŽFig. 1e. showing photon antibunching and finally, they approach again zero value corresponding to the coherent state. As is pointed out in w14,15x, we can also determine an effective nonlinearity parameter as heff s g <1 q g <Ž< a 1 < 2 q < a 2 < 2 .rk , so that in the limit heff 0 the system becomes linear and there occurs only classical switching, the form of which is not affected by the nonlinear coupling, similarly as, for instance, when g s y1 or g < k for rather low intensity of the initial coherent state. The presence of strong nonlinearity heff ` Ži.e., k 0. leads to the selftrapping effect as a consequence of the occurrence of two decoupled Kerr nonlinear waveguides.
™
™
™
4. Conclusions
Fig. 1. Quantum statistical characteristics in the contradirectional Kerr coupler as functions of the length L with the coupling constants k s1, g s 0.25, g s1, for input two-mode coherent state < a 1 , a 2 :, a 1 s 2.5, a 2 s 0. Ža. Evolution of the mean number of photons n1Ž L. Žsolid curve. and n 2 Ž L. Ždotted curve.; Žb. 2 two-mode quadrature variances ² Ž D Xˆ1 . : Žsolid curve. and 2 ² Ž D Xˆ2 . : Ždotted curve.; Žc. two-mode principle squeeze variance lŽ L.; Žd. two-mode uncertainty product uŽ L.; Že. correlations ² D aˆ†1 Laˆ1 L D aˆ†2 0 aˆ 2 0 : Žsolid curve. of integrated intensity fluctuations and fluctuations ²Ž DW1 L,2 0 . 2 : Ždotted curve. of the integrated intensity.
We have investigated the quantum contradirectional coupler composed of two Kerr nonlinear waveguides. We have found approximate solutions of intensity-dependent operator equations. Using operator substitutions, we have obtained Eqs. Ž2. containing linear and nonlinear coupling terms and external higher-order nonlinear source terms. The single-mode solution of the exact equations may be approximated by a solution of weakly coupled equations, whereas the higher-order nonlinear coupling of the other mode can be neglected. This enables us to extract at least main quantum features of the contradirectional Kerr coupler. In fact, a similar idea has been adopted, for instance, in w15,18x as rotating wave approximation. In doing so, we argue that even for contradirectional couplers, this aspect does still work as well. Moreover, the formulation w13x that the operator solutions are written for the input and output of the coupler Žnot inside of the volume., automatically ensures the conservation of the boson commutation rules for all operators involved in Ž1..
G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
Then, when we find a constant of motion of equations, it is constant for all couplers with different lengths. A sufficiently short coupler, approximate solutions of which are known, greatly simplifies the whole problem in such a way that its operators commute with the operators of couplers with different lengths, i.e., it plays a role as a ’reference’ device. Using the quantum consistent solutions, we have calculated the mean number of photons, fluctuations of the integrated intensity and their correlation in single modes and have demonstrated the quantum effect of collapse during switching of the modes, accompanied by enhanced squeezed quantum fluctuations. Two-mode quadrature squeezing and subPoissonian photon statistics are also obtained in this contradirectional device.
Acknowledgements
153
where
L "s
2 l " sinhh 2h coshh y l3 sinhh
ž
L3 s coshh y
, y2
l3 2h
sinhh
/
,
h 2 s 14 l23 q lq ly , we have `
n1 s 4 < a 1 < 2
n Ý Ž y1. Ž n q 1. ey2 k LŽ nq1.
ns0
ˆ ˆ ˆ =² a 1 ,0
here the coefficients read h s gL <1 q g <Ž n q 1., < L < s tanhh , L 3 s sech2h. Using the fact that Jˆ3 z s constant, its exponent may be put on the left-hand side changing its sub-index and returning again to z Žremember that aˆ1 ,2 , aˆ1,2 ., it holds that z z `
This work was supported by Complex Grant No VS96028 and Research Project CEZ: J14r98 of the Czech Ministry of Education.
n1 , 4 < a 1 < 2
n Ý Ž y1.
ns0
= Ž n q 1 . ey2 k LŽ nq1. eŽexpŽln'L 3 .y1.< a 1 < )
=² a 1 ,0
We numerically calculate the mean number of photons Ž6. expanding the function f Ž x . s sech2 r s 4rŽ x q 1rx q 2. with x s ey2 r Ž r G 0. into Taylor series as ` n Ý Ž y1. Ž n q 1.
ns0
ˆ =² a 1 ,0
Ž A.1 .
If we use the ordering theorem for the angular momentum operators w19x exp lq Jˆqq ly Jˆyq l3 Jˆ3
ž
†
e L aˆ 2 a 1 < a 1 ,0: .
Ž A.3 .
Using the Baker–Hausdorff formula for the second mode, we obtain the expression Ž7..
Appendix A
n1 s 4 < a 1 < 2
aˆ 2
2
/
s exp Ž Lq Jˆq . exp Ž ln L3 . Jˆ3 exp Ž Ly Jˆy . ,
Ž A.2 .
References w1x A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984. w2x J. Janszky, C. Sibilia, M. Bertolotti, J. Mod. Opt. 38 Ž1991. 2467. w3x M. Toren, Y. Ben-Aryeh, Quantum Opt. 6 Ž1994. 425. w4x J. Perina, ˇ Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd edn., Kluwer, Dordrecht, 1991. w5x W.K. Lai, V. Buzek, ˇ Knight, Phys. Rev. A 43 Ž1991. 6323. w6x G. Assanto, A. Laureti-Palma, C. Sibilia, M. Bertolotti, Opt. Commun. 110 Ž1994. 599. w7x J. Perina, J. Perina ˇ ˇ Jr., Quantum Semiclass. Opt. 7 Ž1995. 849. w8x A. Luis, Perina, Quantum Semiclass. Opt. 8 Ž1996. 39. ˇ w9x J. Perina, Acta Phys. Slov. 45 Ž1995. 279. ˇ w10x J. Perina, J. Perina ˇ ˇ Jr., Quantum Semiclass. Opt. 7 Ž1995. 863. w11x N. Korolkova, J. Perina, Opt. Commun. 137 Ž1997. 263. ˇ ˇ ˇ L. Mista w12x J. Rehacek, J. Mod. Opt. 46 Ž1999. ˇ Jr., J. Perina, ˇ 801.
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G. Ariunbold, J. Perinar Optics Communications 176 (2000) 149–154 ˇ
w13x J. Perina Progress in Optics, Vol. 41, E. Wolf ˇ Jr., J. Perina, ˇ ŽEd.., Elsevier, Amsterdam, in print. w14x A. Chefles, S.M. Barnett, J. Mod. Opt. 43 Ž1996. 709; J. Fiurasek, J. Perina, Opt. Commun. 167 Ž1999. ´ˇ J. Krepelka, ˇ ˇ 115. w15x N. Korolkova, J. Perina, Opt. Commun. 136 Ž1996. 135; J. ˇ Mod. Opt. 44 Ž1997. 1525. w16x R.W. Boyd, Nonlinear Optics, Academic Press, Boston, 1992, chapter 4.
w17x R. Tanas, ´ Coherence and Quantum Optics 5, L. Mandel, E. Wolf ŽEds.., Plenum, New York, 1983, p. 645. w18x A. Mecozzi, S. Trillo, S. Wabnitz, B. Daino, Opt. Lett. 4 Ž1987. 275; A.P. Alodjants, S.M. Arakelian, A.S. Chirkin, Quantum Semiclass. Opt. 9 Ž1997. 311. w19x M. Ban, J. Math. Phys. 33 Ž1992. 3213; J. Opt. Soc. Am. B 10 Ž1993. 1347.