Quantum Phase Space Kerr Oscillator

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15 August 1999

Ž .Optics Communications 167 1999 115–124

www.elsevier.comrlocateroptcom

Quantum-phase properties of the Kerr couplers

J. Fiurasek a, J. Krepelka b, J. Perina a,b,)´ˇ ˇ ˇa

Department of Optics, Palacky UniÕersity, 17 Listopadu 50, 772 07 Olomouc, Czech Republic´b

Joint Laboratory of Optics, Palacky UniÕersity and Institute of Physics of Academy of Sciences of the Czech Republic, 17 Listopadu 50,´772 07 Olomouc, Czech Republic

Received 2 March 1999; accepted 3 June 1999

Abstract

We use the concept of phase space and Husimi quasidistribution to derive joint-phase probability distribution andquantum-phase properties for the Kerr couplers. The exact numerical as well as approximate analytical solutions of the

Schrodinger equation are found. The spatial development of the single-mode phase distributions and phase-difference¨distribution is demonstrated. The Fourier coefficients of the phase distributions are introduced and employed to describe

quantum-phase behaviour. It is shown that the phase-difference evolution is closely connected to an energy exchange

between two waveguides, which form the coupler. The collapses and revivals of the mean photon number oscillations are

due to the bifurcation of the phase-difference probability distribution, which has a two-fold symmetry in the interval of

collapse. q1999 Elsevier Science B.V. All rights reserved.

PACS: 42.50.Dv

Keywords: Quantum phase; Non-linear coupler; Kerr effect

1. Introduction

The aim of this paper is to study phase propertiesof the optical modes propagating in the Kerr non-lin-

ear coupler. This non-linear optical device consistsof two parallel waveguides whose modes are mutu-ally linearly coupled and each mode is non-linearly

coupled to itself by means of the Kerr effect. Wealso include the non-linear cross-interaction. The

)Corresponding author. Fax: 420-68-5225246; e-mail:

[email protected]

Kerr coupler can be described with the help of thew xinteraction momentum operator 1,2

ˆ ˆ† † ˆ †2 2 ˆ† 2 ˆ 2G s"k ab q a b q"ga a q"gb bˆ ˆ ˆ ̂Ž .int

† ˆ† ˆq"ga ab b . 1Ž .˜ˆ ˆ

† ˆ ˆ†Ž . Ž . Ž .Here a a and b b are annihilation creationˆ ˆoperators of the modes in the first and second wave-guides in the interaction picture, k is the linear

coupling constant between the two modes, g and g̃

are non-linear coupling constants proportional to thethird-order susceptibility characterizing the strengthof the self-action and cross-action processes, respec-

0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .P I I : S 0 0 3 0 - 4 0 1 8 9 9 0 0 2 8 6 - 2



( ) J. Fiurasek et al.r Optics Communications 167 1999 115–124´ˇ116

tively. It is assumed that the propagation constants of both modes are the same.

Analysis of such a device, based on a classicalwave optical description, reveals that the couplerexhibits several different regimes of operation independence on the values of linear and non-linear

w xcoupling constants and input intensities 3– 5 . Alsothe soliton propagation in the Kerr couplers has been

investigated and regimes of bifurcations were foundw xand discussed 6,7 . In standard considerations theenergy is injected into the first waveguide and itstransmission to the second waveguide is analyzed. A

certain threshold value of the input intensity exists;the energy is periodically fully exchanged betweenthe waveguides below the threshold; the intensities

are asymptotically uniformly distributed at thethreshold; and, finally, when the input intensity ex-ceeds the threshold value, the non-linear effect locks

the energy in the waveguide, where it was initiallyinjected, and the exchange of the energy decreases.

The equivalent quantum model has been dis-w xcussed in Ref. 1 . The Schrodinger equation in the¨

basis of Fock states was solved numerically and theevolution of mean numbers of photons for initialFock and coherent states was studied. The quantum

Kerr coupler behaves quite differently from its clas-sical counterpart for small input intensities. Particu-larly, collapses and revivals of the photon number

oscillations take place if the coupler operates belowthe threshold. To explain such a behaviour, an ap-proximate analytical solution was found for the ini-

tial Fock state. This solution is based on a transfor-Žmation of the basis new annihilation operators de-

fined as linear combination of the old ones are.introduced and neglection of the part of the momen-

tum operator leading to its diagonalization in thenew basis.

w xThis idea was fully unfolded in Ref. 2 , wherethe evolution of the annihilation operators in theHeisenberg picture was obtained in analytical ap-

proximation. Assuming initial coherent states, theanalytical expressions for mean photon numbers,principal squeeze variance and integrated intensity

variance were derived and a possibility of generationof non-classical states of light was discussed. Theinfluence of the z-dependence of the linear coupling

constant on the switching properties of the couplerwas also investigated within the framework of these

w xapproximate analytical expressions 8 . The analyti-cal results give us important general informationabout the behaviour of the coupler. However, these

predictions fail for long distance z or high values of non-linear coupling constants. In those cases, thenumerical calculation has to be employed.

In contrast to the classical description, the transi-tion from operation below the threshold to the

above-threshold regime is smoothed in the quantummodel when only small numbers of input photons areassumed. The quantum dynamics of the coupler isdetermined by a countable set of the eigenvalues of

the momentum operator and effectively only a finitenumber of them contributes substantially to the evo-

< :lution of the state vector C . Nevertheless, thenotion of the threshold is still useful because, with

increasing g, the behaviour of the coupler substan-w xtially changes 1 . One can use classical equations to

w xdetermine the threshold 3 . When all energy isŽ .initially injected into the first waveguide b s 0 ,

then the operation of the coupler depends on an<effective non-linearity parameter h s 2 g y

< < < 2g a rk . The classical threshold is reached at h s 4.˜We are below the threshold for h - 4 and above thethreshold for h ) 4. Notice that for g s 2 g the˜parameter h s 0 regardless of the value of a and k

and the coupler behaves like a linear one. We havefixed a s 2, b s 0 and k s 1 in our calculations andchanged g and g to obtain various regimes of the˜coupler operation.

In this paper we investigate the phase propertiesof the modes propagating in the coupler. The theory

of quantum phase has attracted much attention dur-ing past years when several approaches to the quan-

w x Žtum phase were proposed 9–12 for a review, seew x.Ref. 13 . Quantum-phase properties of elliptically

polarized light propagating in a Kerr medium havebeen investigated within the framework of the

w xPegg–Barnett Hermitian phase formalism 14–16 .In the present work we use the concept of a phase-

w xspace approach 17 , which has been recently appliedto the investigation of phase properties of contradi-

rectional non-linear coupler operating by means of

w xdegenerate parametric down-conversion 18 . WeŽ .calculate the phase distribution P w from theŽ ) .Husimi quasidistribution F a , a related to the

A A

antinormal ordering of field operators. We show thatŽ .the function P w can give us an important informa-



( ) J. Fiurasek et al.r Optics Communications 167 1999 115–124´ˇ 117

tion about the quantum state of the mode and about

the behaviour of the quasidistribution. Particularly,several peaks of this function signalize quite com-plex form of the quasidistributions having also sev-

eral peaks, as will be shown below.

2. Phase-space approach to the quantum phase

2.1. Phase distributions

The single-mode Husimi quasidistribution may be

expressed in terms of single-mode density matrix r ,ˆ

` ) m n1 r a a 2m n) y < a <F a , a s e . 2Ž . Ž .Ý A A 'p m! n!m, ns0

² < < :Here r s m r n are density matrix elements inˆm n

Fock basis. The F function is a real non-negative A A

function normalized to the unity. These propertiesare transferred to the phase distribution. The defini-

Ž . w xtion of P w is 17

`

i w yi w < < < < < < < <P w s F a e , a e a d a , 3Ž . Ž .Ž .H A A

0

Ž . 2p Ž .and it holds that P w 0 0 and H P w d w s 1.0

Ž . Ž .The substitution of 2 into 3 and the integration< <over a yield

m q nG q 1`

1 ž /2 iŽ nym.w P w s r e .Ž . Ý m n '2p m! n!m, ns0

4Ž .

Ž .In order to calculate P w we have to determine theŽ . z-dependent density matrix r z of the single modes.ˆ

We can also introduce the two-mode joint-phase

probability distribution, defined in terms of two-modeF -function A A

` `

) )P w , w s F a ,a ;a , aŽ . Ž .H H1 2 A A 1 1 2 20 0

=< < < < < < < <a a d a d a , 5Ž .1 2 1 2

< < Ž .where a s a exp i w , j s 1,2. The two-mode j j j

Husimi quasidistribution can be expressed with the

help of two-mode density matrix elements r m m , n n1 2 1 2

² < < :s m , m r n , n similarly as the single-modeˆ1 2 1 2

Ž .quasidistribution 2 . The direct integration leads to

` `1P w , w s r Ž . Ý Ý1 2 m m , n n2 1 2 1 22pŽ . m , n s0 m , n s01 1 2 2

=

m q n m q n1 1 2 2G q 1 G q 1

ž / ž /2 21r2

m ! n ! m ! n !Ž .1 1 2 2

=eiŽ n1ym 1.w 1 eiŽ n 2ym 2 .w 2 . 6Ž .

Further we define phase difference Dw s w y w 1 2

whose probability distribution can be easily deter-mined,

2p 2pP Dw s P w , w Ž . Ž .H H 1 2

0 0

=d Dw

yw

qw

dw

dw

, 7Ž . Ž .1 2 1 2

where d is the Dirac delta function. Hence we have

P Dw Ž .

` `1s r Ý Ý m m , n Ž m qm yn .1 2 1 2 1 12p

m , n s0 Ž .m smax 0, n ym1 1 2 1 1

=

m q n 2 m q m y n1 1 2 1 1G q 1 G q 1ž / ž /2 2

1r2m ! n ! m ! m q m y n !Ž .1 1 2 2 1 1

=eiŽ n1ym 1 .Dw . 8Ž .

2.2. Phase dispersions and Fourier coefficients

We conclude this section with some remarks aboutthe phase-distribution variance. The usual definition

2 ²Ž ² :.2 :s s w y w is not suitable for our purposes,Ž .because P w is a periodic function and the variance

depends on the choice of the integration interval.

Thus the so-called phase dispersion has been intro-w xduced in Ref. 19 , using the mean values of periodicfunctions sin w and cos w ,

2 ² :2 ² :2 <² i w : < 2s s 1 y sin w y cos w s 1 y e . 9Ž .




This expression is suitable for calculations. The mean² Ž .:value exp i w can be calculated with the help of

Ž .the general formula 4 for the phase distributionŽ .P w , yielding

3` r G n qŽ .nq1 , n 2i w ² :e s . 10Ž .Ý 'n! n q 1ns0

Ž .Using 9 we can easily interpret the meaning of the<² Ž .: <phase dispersion. The quantity exp iw represents

an amplitude of one Fourier coefficient in the spec-

tral Fourier decomposition of the phase-probabilitydistribution. Thus the quantity s provides informa-tion about the appearance of the fundamental fre-

quency in the spectral decomposition of the phase-Ž .probability distribution P w . The fundamental com-Ž .ponent induces one peak maximum of the function

Ž .P w . Similarly the higher harmonics support n

peaks of the probability distribution. We can general-ize the definition of s and introduce higher-order

Ž .dispersions s k ,

2 <² i k w : < 2s k s 1 y e , k g N . 11Ž . Ž .

They allow us to study the appearance or disappear-Ž . Ž ) .ance of n-fold symmetry in P w and F a , a

A A

Ž .functions. When s n decreases the n-fold modula-tion of the distributions is more pronounced and vice

versa. Therefore it is useful to study directly thespatial or temporal development of the Fourier coef-

Ž . <² Ž .: <ficients F k s exp i k w . A simple generaliza-Ž .tion of the expression 10 yields

2 n q k r G q 1

` nq k , n ž /2F k s . 12Ž . Ž .Ý 1r2

n! n q k !Ž .ns0

The Fourier coefficients of the phase-difference dis-tribution can be introduced in a similar way. A closerŽ . Ž .inspection of the formulas 4 and 8 reveals that the

expressions for the single-mode phase distributionŽ . Ž .P w and phase-difference distribution P Dw are

formally equivalent, if we define an effective densitymatrix for the latter case,

2 s q m y nG q 1

` ž /2r s r ,˜ Ým n m s , nŽ sqmyn. 1r2

s! s q m y n !Ž .ss0

m 0 n ;

r s r )

, m - n . 13Ž .˜ ˜m n n m

Ž .This effective density matrix may be inserted in 12 ,which yields the required Fourier coefficients of thephase-difference distribution.

3. Numerical and analytical solutions of the

Schrodinger equation¨

We assume only pure states. The state vector

< Ž .:C z is a solution of the Schrodinger equation,¨d

ˆ< : < :yi" C z s G C z 14Ž . Ž . Ž .d z

and may be expressed in the basis of the Fock states< : < : < :n ,n ' n n ,1 2 1 2

`

< : < :C z s c z n , n ,Ž . Ž .Ý n n 1 21 2

n , n s01 2

² < :c z s n ,n C z . 15Ž . Ž . Ž .n n 1 21 2

< : ² <The density operator can be written as r s C C ˆŽ .and its matrix elements are r z sm m , n n1 2 1 2

Ž . ) Ž .c z c z .m m n n1 2 1 2

Ž . Ž .The solution 15 of Eq. 14 can be easily foundw xnumerically. This was described in detail in Ref. 1

and we point out only the most important facts. TheŽ .coefficients c z have to be calculated. Then n1 2

Hilbert space of all states splits into a direct sum of invariant subspaces due to the conservation of the

† ˆ† ˆŽtotal number of photons operator n s a a q b bˆ ˆ ˆ.commutes with the momentum operator ,

H H s [`

H H , 16Ž .ns0 n

the dimension of each subspace HH being n q 1 andn

< :the basis can be chosen in the form j,n y j , j s0 , . . . ,n. The Schrodinger equation is rewritten into¨




matrix form for each subspace, the eigenvalues ln j

< :and eigenvectors C of the corresponding matricesn j

G ,n

1X X XˆX ² < < :G s j, n y j G j , n y j , j, j s 0 , . . . ,n ,Ž . j jn

"

17Ž .

are found and the initial conditions are applied toŽ .obtain the coefficients c z . Defining vectorsn n1 2

T T< :C z s c z ' c z , . . . ,c z ,Ž . Ž . Ž . Ž .n j , ny j 0 , n n ,0

18Ž .

we can write

n

i l zn j< : ² < : < :C z s e C C 0 C . 19Ž . Ž . Ž .Ýn n j n n j

js0

If initial coherent states are assumed with complex

amplitudes a , b , then

n1 n 2 < < 2 < < 2a b a q b

c 0 s exp y . 20Ž . Ž .n n1 2 ž /2n ! n !( 1 2

The density matrices of single modes can beexpressed as traces over the other mode,

`

)r z s c z c z ,Ž . Ž . Ž .Ý1 , m n m r n r

r s0

`

)r z s c z c z . 21Ž . Ž . Ž . Ž .Ý2 , m n r m r n

r s0

We can also employ the approximate analyticalsolutions to obtain the analytical expressions for the

Ž . w xcoefficients c z . Following Ref. 1 we introducen n1 2

new operators

1 1ˆ ˆ ˆ ˆ A s a q b , B s a y b . 22Ž .Ž . Ž .ˆ ˆ' '2 2

These new operators obey the same commutationˆrelations as a and b. We define Fock states in termsˆ

of them

< : † j † N y j < : j, N y j s j A B 0 ,0 , N j

1j s . 23Ž . N j ( j! N y j !Ž .

< :These states differ from the Fock states n , n . To1 2

clarify the notation, we use N for the states definedŽ . Ž .in 23 . An inversion of the definition 22 and

Ž .substitution into 1 give the momentum operatorˆ ˆexpressed in terms of the operators A, B,

ˆ ˆ† ˆ ˆ† ˆG s"k A A y B BŽ .

" †2 2 † 2 2ˆ ˆ ˆ ˆq 2 g q g A A q B BŽ .˜ Ž .4

ˆ† ˆ ˆ† ˆq 2"gA AB B

"†2 2 †2 2ˆ ˆ ˆ ˆq 2 g y g A B q B A . 24Ž . Ž .˜ Ž .

4

To obtain the analytical results we omit the rotatingˆ†2 ˆ2 ˆ†2 ˆ2term proportional to A B q B A in the momen-

Ž .tum operator 24 . This approximation is equivalentto the neglection of corresponding rotating terms in

ˆ ˆthe Heisenberg equations for the operators A, B,w xwhich was done in Ref. 2 and the two approaches

give identical results.The neglection of the above two terms leads to

ˆthe diagonalization of the momentum operator G in< :the basis j, N y j , corresponding eigenvalues being

1L s k 2 j y N q 2 g q g N N y 1Ž . Ž . Ž .˜ N j 4

1q 2 g y g j N y j . 25Ž . Ž . Ž .˜2

Now we can write the solution in this new basis and

transform it back to the former basis to obtain theŽ .coefficients c z . We haven n1 2

` N i L z N j< : < :C z s e d j, N y j ,Ž . Ý Ý N j

N s0 js0

² < :d s j, N y j C 0 . 26Ž . Ž . N j

We assume initial coherent states in the following< Ž .: < :considerations, i.e. C 0 s a , b . With the help of

Ž .the definition 23 we arrive at

j N y jy N r2d s j 2 a q b a y bŽ . Ž . N j N j

=< < 2 < < 2a q b

exp y . 27Ž .ž /2




A straightforward calculation based on the definitionŽ . Ž .of c z in Eq. 15 yieldsn n1 2

n ! n !1 2c z sŽ . (n n n qn1 2 1 22

=

n qn1 2

n y j2exp i L z y1Ž .Ž .Ý n qn , j1 2

js0

=j d R ,n qn , j n qn , j n n j1 2 1 2 1 2

Ž .min j, n1 n q n y j j 1 2k R s y1 .Ž .Ýn n j1 2 ž / ž /n y k k 1Ž .k smax 0, jyn 2

28Ž .

Of course, we cannot sum over the whole Hilbert

space, but we must choose the upper limit of thesum, n q n s N . The same holds for numerical1 2 max

solutions discussed above. We must choose N somax

that the Hilbert subspace is large enough to contain

all important information about the process underdiscussion. The upper limit depends significantly on

² :the total mean number of photons n s n , for ourˆcalculations n s 4 and N s 20 has proved to bemax

sufficient. Both numerical and analytical calculationspermit the independent tests of their validity because

a mean photon number in each Hilbert subspace HH n

should be conserved during the evolution.

4. Discussion

The same initial conditions were chosen in Refs.w x1,2 to investigate a spatial evolution of the mean

photon numbers and this allows us to use the resultsobtained there. In the first part of the discussion weanalyze the spatial evolution of the single-mode phase

distributions and in the second part we focus our-selves on the spatial evolution of the phase differ-ence Dw . We show that the latter is closely associ-

ated with the spatial development of the mean pho-ton numbers and that the phase distributions canshed some light on the processes arising from simul-taneous influence of linear coupling and non-linear

self- and cross-phase modulations. The discussion is

based on numerical results. The approximate formu-

las are generally useful only for small non-linearitiesand short distances z. However, they become exactwhen g s 2 g and this special case will be discussed˜in detail.

4.1. Single-mode phase distributions

We will confine ourselves to the first mode be-cause the behaviour of the second mode is similar.

To investigate spatial evolution of the distributionŽ . Ž .P w , we can employ Fourier coefficients F k , z1 1

Ž .as a powerful tool subscript refers to the first mode .

They are in close relation with the phase dispersions2 Ž . Ž . Ž .s k according to 11 ; if a function F k , z de-

2 Ž .creases, then the corresponding dispersion s k

increases and vice versa. It is sufficient to deal with

the first four or five coefficients which comprisealmost all information about the phase distribution.

< : Ž .For the coherent state j , P w has one peak atŽ . Ž .w s arg j and the coefficient F 1, z prevails.1

Ž .The spatial development of F k , z for two dif-1

ferent values of non-linear parameters is shown in

Fig. 1. It is useful to remember the results obtainedw xfor the light propagation in Kerr media 14–16 . If

two polarization modes are taken into account, the

description is equivalent to our model without linearcoupling, i.e. with k s 0. The analytical results,obtained in this case, reveal that the spatial evolution

Žcan be periodic this takes place if grg can be˜.expressed as a fraction of integers and that the state

of the field becomes a discrete superposition of finite

number of coherent states for t s MT r N , T is aperiod and M , N are primes. The appearance of N -fold structure can be observed in Fig. 1 and it

Ž .corresponds to peaks of F N , z . Simultaneously N 1

Ž .peaks also appear in the phase distribution P w 1

Ž ) . Žand quasidistribution F a , a see Figs. 2 and A A ,1

.3 . It is obvious from Fig. 1a that the linear coupling

constant k breaks the periodicity down. The peak of Ž .F 1, z in Fig. 1a at z f 20p is connected with the1

revival of the energy exchange between the wave-

guides which will be discussed in greater detail inthe next subsection. The structure of joint-phaseŽ .distribution P w ,w can be quite complex. Particu-1 2

larly the N -fold symmetry is exhibited there clearly.The total number of peaks may differ from N when




Ž .Fig. 1. Spatial evolution of the Fourier coefficients F k , z of the1

first-mode phase distribution. The linear coupling constant k s1;Ž . Ž .a g s0.1, g s0; b g s0.1, g s0.2. The first mode is˜ ˜initially in the coherent state with complex amplitude a s2 and

the second mode is initially in vacuum state, b s0.

2 w xg s2 g to N when g s 0 15 . An example is˜ ˜given in Fig. 4. The distance z s 20.7 was chosen to

Ž .coincide with the peak of F 2, z in Fig. 1a. Indeed,1the two-fold structure is present. Note that due to the

Ž .influence of linear coupling the function P w ,w 1 2

is not completely symmetric.In a special case when g s 2 g the periodic be-˜

haviour of the spatial evolution can be restored, if a

relative value grk can be expressed as a quotient of two integers. This is shown in Figs. 1 and 2. We cansee the evolution arising from the non-linear phase

modulation with the period Z sprg, which is su-perimposed with oscillations induced by k . Thisholds even if grk is not a ratio of two integers.

Then beats between the two oscillations having peri-ods Z sprg and Z sprk appear and the initial1 2

state is never fully reconstructed. It should be pointed

out that in this case the energy exchange between thewaveguides is equivalent to the linear waveguide, the

Fig. 2. Spatial evolution of the phase-probability distributionŽ .P w , z of the first mode; the parameters are the same as in Fig.1

1b.

energy is entirely periodically transferred betweenw xthe waveguides with the period Z 2 . This follows2

from the analytical results presented in the previoussection, which are exactly valid for g s 2 g. Notice˜

Ž . Ž .that F k , z has k peaks in the interval 0, Z .1 1

However, some peaks are split into two shiftedpeaks. This is a consequence of the energy exchange

Ž ) .Fig. 3. QuasidistributionF a

,a

for the first mode; the AA ,1

parameters are the same as in Fig. 2 and z s10. This choice of z

Ž .corresponds to the peak of F 3, z in Fig. 1b and it slightly differs1

Ž .from the value pr 3 g s10pr3. This shift is induced by linear² Ž .: ² Ž .:coupling. It holds that n 10 ) n 10pr3 and thus the1 1

3-fold modulation is more pronounced for z s10.




Ž . Ž .Fig. 4. Phase distribution P w ,w for z s20.7. a 3-D view.1 2Ž .b Topographic plot. The parameters are the same as in Fig. 1a.

between the waveguides. The peaks are most pro-nounced if the amplitude is maximal, the missing

Žpeaks correspond to the zero amplitude vacuum.state of the first mode. If g ) 2 g, then the periodic-˜

ity of the spatial evolution disappears again and thebehaviour is similar to g - 2 g.˜

4.2. Phase-difference distribution

The relative phase Dw whose distribution is de-

Ž .fined by 8 , is closely associated with a spatialbehaviour of the mean photon numbers in two modesof the coupler. The condition for an optimal energy

exchange between the waveguides is that the phasedifference between the complex amplitudes of the

first and second modes is "pr2. The functionŽ .P Dw reflects it as a peak at pr2 or 3pr2.

Ž .The spatial development of P Dw can again beextracted using appropriate Fourier coefficients. Theirspatial evolution for two different cases is shown inFig. 5. In Fig. 5a we can see the spatial evolution of

Ž .the Fourier coefficients F k , z when only self-phasemodulation is present and g s 0. It is obvious that˜

Ž .with increasing z the coefficient F 2, z becomesdominant and the two-fold symmetry of the phase-difference distribution is established. This bifurcationimplies the appearance of collapse of the energy

² :exchange between the waveguides when n fˆ1

² : ² : w xn f n r2 for a quite long interval of z 1,2 .ˆ ˆ2

Notice that a similar bifurcation of single-mode phase

distribution has been obtained for highly squeezedw xstates 20,18 . Finally, the revival takes place and the

energy exchange is re-established. However, it is

weaker and only a part of the energy is periodicallyexchanged.

Ž .Fig. 5. Spatial evolution of the Fourier coefficients F k , z of the

phase-difference distribution. All parameters are the same as in

Fig. 1a,b.




Ž .The phase-difference distribution P Dw can be

seen in Fig. 6. For z s 0 it is just a straight line,Ž .P Dw s 1r2p, because the second mode is ini-

tially in a vacuum state. For z spr4 the distribution

has one peak at Dw s 3pr2 and for z s 3pr4 ithas one peak at Dw spr2, which corresponds tothe almost complete periodic energy exchange for

small z; z s 10p holds in the region of the collapse

of the oscillations. The distribution has two-foldsymmetry, the position of the peaks being pr2 and3pr2. We move to the revival area for z s 20p. We

Fig. 6. 3-D plot of a spatial evolution of the phase-differenceŽ . Ž . Ž . Ž .distribution P Dw , z a and 2-D plot of P Dw for z s0 — ,

Ž . Ž . Ž . Ž . z spr4 ` , z s3pr4 I , z s10p ^ and z s20p e

Ž .b . The parameters are the same as in Fig. 1a.

can see that the distribution has now only one peak

but it is flatter, because the revival is only partial.The increase of g towards the threshold value

leads to a qualitative change of the spatial develop-

ment of all the assumed quantities. The coefficientŽ .F 1, z becomes dominant for all z. Thus no col-

lapses and revivals appear but oscillations take place.

With a further increase of g above the threshold the

energy becomes trapped in the waveguide where itwas initially injected and thus the amplitude of the

energy exchange decreases. This is reflected in thephase-difference distribution, with increasing g the

Ž .average values of F k , z decrease and tend to zero,

which signalizes the flattering of phase-differencedistribution. This behaviour takes place not onlywhen g - 2 g but also for g ) 2 g.˜ ˜

ŽThe case g s 2 g regardless of the absolute value˜.of these constants forms an exception as it can be

Ž .seen in Fig. 5b. The evolution of P Dw is periodic,the period being Z sprk. As a matter of fact the2

phase difference behaves in the same way as if thecoupler was linear and the two modes were in thecoherent states. This result directly follows from the

analytical solution. If g s 2 g, then the momentum˜Ž .operator eigenvalues L defined in 25 are simpli- N j

fied and we have

L s k 2 j y N q g N 2 y N . 29Ž . Ž . Ž . N j

This expression is precise. The first term in the sumcorresponds to the linear coupler. The second term,which is a function of N only, results in a non-linear

phase modulation of the single modes but is can-Ž . Ž .celled in the formula 8 for P Dw .

5. Conclusions

We have investigated quantum-phase propertiesof optical fields propagating in the Kerr couplers.We have employed the phase-space approach work-

ing with the Husimi quasidistribution related to anti-

normal ordering of field operators. The numericaland approximate analytical solutions of the corre-

sponding Schrodinger equation, describing quantum¨dynamics of the coupler, were found. Fourier coeffi-cients of the phase-probability distribution were in-




troduced and utilized to examine the spatial develop-ment of the phase distributions. It was shown that the

mutual interaction of linear coupling and non-linearphase modulation destroy the periodicity of the spa-tial development, which can take place for k s 0.However, such a periodicity is re-established if non-

linear self- and cross-interactions compensate eachother, i.e. g s 2 g.˜

Ž .The peaks of the Fourier coefficients F k , z indi-cate a k -fold symmetry of the phase distribution andthe corresponding single-mode anti-normal quasidis-tribution. The two-fold symmetry of the phase-dif-

Žference distribution i.e. a phase-difference bifurca-.tion is responsible for the collapse of photon-num-

ber oscillations. If the above mentioned compensa-

tion of non-linear interactions takes place, the phasedifference behaves like that of a linear coupler withthe same coupling constant k and input coherent

states. Thus the energy is periodically exchangedbetween the two modes of the coupler in this case.

Acknowledgements

This work was partly supported by Grant No.

VS96028 of the Czech Ministry of Education.

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Quantum Phase Space Kerr Oscillator