PHYSICAL REVIEW A VOLUME 48, NUMBER 3 SEPTEMBER 1993

Quantum-mechanical stability of solitons and the correspondence principle

F. X. Kartner and H. A. HausDepartment ofElectrical Engineering and Computer Science and the Research Laboratory ofElectronics,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Received 1 March 1993)

A Hanbury Brown —Twiss experiment is described that is capable of demonstrating, in principle, thesolitary behavior of a quantum soliton in an arbitrary fundamental soliton state. The relevant correla-tion functions are evaluated and show that the quantum noise does not cause a breakup of the solitons ofthe quantized nonlinear Schrodinger equation. For large photon numbers, the intensity autocorrelationfunction of the fundamental soliton becomes independent of certain momentum and photon-number dis-

tributions, which gives us a correspondence principle for the fundamental soliton. Whereas intensity-

correlation measurements based on direct detection show essentially classical behavior for large photonnumbers, homodyne-detection experiments can uncover quantum e8ects such as the generation ofSchrodinger-cat solitons.

PACS number{s): 42.50.Rh, 42.50.Lc, 42.65.—k

I. INTRODUCTION

Recently, optical-field propagation in a nonlinear fiberhas been studied extensively in a number of papers [1—9].The motivation is to gain understanding of the squeezingeffect in soliton propagation [1—7], quantum revivalsoccurring in the absence of dissipation [8], and the funda-mental limits imposed on communication systems em-ploying solitons. These fundamental limits are deter-mined by quantum effects such as the Gordon-Haus effect

The classical soliton is governed by the classical non-linear Schrodinger equation (CNSE), and one expectsfrom the correspondence principle that the quantum non-linear Schrodinger equation (QNSE) is the proper equa-tion of motion for the corresponding quantized field.Similarly to the CNSE [10], the QNSE has also beensolved analytically [11—16]. One way to obtain analyticalsolutions to the QNSE is the use of the Bethe ansatzmethod [17] to solve for the eigenstates and energy eigen-values of the QNSE [12]. If the coefficient of the non-linear term in the QNSE is negative, there exist boundeigensolutions to the corresponding Hamiltonian. Fol-lowing the work of Nohl [15] and Wadati and Sakagami[18,19], Lai and Haus showed that only a superpositionof a subclass of these bound states which are character-ized by the photon number and the momentum of thecenter coordinate of the bound photons can reproducethe expectation values of the field of a classical soliton[5,6] with a well-defined amplitude, phase, position, andmomentum. The momentum of the soliton is related tothe center frequency of the optical pulse. In the case of aclassical soliton, the amplitude of the soliton and its tem-poral or spectral width are closely related since the soli-ton phenomenon in a fiber occurs due to the balance be-tween dispersion and self-phase modulation. The largerthe bandwidth of the soliton, the more the pulse is subjectto dispersion; the shorter the soliton, the larger its ampli-tude so as to increase the self-phase modulation which

balances the dispersion. Therefore, it is not obvious thatthe general quantum state of arbitrary photon statisticsand arbitrary distribution of momentum, i.e., center fre-quency, as stated by Lai and Haus [6] shows solitarybehavior over arbitrary propagation distances. It is notdifficult to derive the fact that the first-order (fundamen-tal) soliton disperses so that the probability of detectingthe soliton at a specified position approaches zero as thetime of propagation approaches infinity [5,20]. However,this fact does not prove, or disprove, the stability of thequantum soliton. It can be shown, using the formalism ofRef. [6] that the intensity-correlation function of a quan-tum soliton is time independent [21].

In this paper we describe a form of a HanburyBrown —Twiss experiment [22] that demonstrates the sta-bility of the fundamental soliton, and we develop expres-sions for the hierarchy of correlation functions measuredin such an experiment. In Sec. II we briefly review thequantum theory of soliton propagation in theSchrodinger picture. In Sec. III we design and analyzethe Hanbury Brown —Twiss experiment which proves thestationary properties of the fundamental soliton statewith respect to autocorrelation measurements of the in-tensity. Whereas these autocorrelation experimentsessentially show classical behavior for arbitrary funda-rnental soliton states with large photon numbers, weshow in Sec. IV that homodyne-detection experimentscan uncover more subtle quantum features. This discus-sion shows that, despite the fact that all first-order soli-tons of high photon number behave classically withrespect to intensity autocorrelation measurements, theydo not behave like classical solitons with respect tohomodyne detection.

II. QUANTUM THEORY OF THE SOLITON

The quantized field in a fiber exhibiting self-phasemodulation due to the instantaneous Kerr effect and

1050-2947/93/48(3)/2361(9)/$06. 00 48 2361 1993 The American Physical Society

2362 F. X. KARTNER AND H. A. HAUS

[P(t,x),P (t,x')]=5(x —x'),[P(t,x),P(t, x')]=[/ (t, x),P (t,x')]=0. (2)

The QNSE is the equation of motion generated by theHamiltonian

H=fif P (t, x)P (t, x)dx

+cf P (t, x)P (t, x)P(t, x)P(t, x)dx, (3)

and in the Schrodinger picture, pulse propagation can bedescribed by

(4)

The general many-particle statei g ) can always be

decomposed into n-particle states according to [23]:

—f d [ xIf„( „x. . . , „x,t)n!

Xy'(x, ), . . . , y'(x„)io), (5)

where the n-particle wave functions f„(x„.. . , x„,t)have to be square-integral functions with norm 1, and thecoefticients a„which determine the photon statistics ofthe pulse have also to be square summable:

group-velocity dispersion is governed by the QNSE

82-i P(t, x) = — P(t, x)+2cg (t, x)P(t, x)(S(t,x) .

Bx

The normalization is identical with that in Ref. [5].Thus, t is the normalized propagation distance z in theAber, and x =z —

U~~ is the time normalized to a distanceby the group-velocity vg in a coordinate system comovingwith the group velocity in the fiber. Therefore, P+(t, x)and P(t, x) are creation and annihilation operators of aphoton at x and t. These field operators obey the usualcommutator relations for bosons:

by only one additional quantum number p besides thephoton number n and are given by

nCf„~=N„exp ip g x.+-

j=l l, J1~i (j~n

ix, —x, i

where X„ is a proper normalization constant,

1/2(n 1)!icl I

2m(9)

so that Eq. (6) is fulfilled. It is obvious from the construc-tion of this wave function that the quantum number p isrelated to the momentum of the center coordinate of then interacting bosons. The wave function decays exponen-tially with the separation distances of boson pairs. Thewave function f„„ is the wave function of an n-photonsoliton moving with momentum p. The energy eigenval-ues of these n-photon solitons are

E(n,p)=np — n(n —1) .12

(10)

The energy consists of the sum of kinetic energy of the n

bosons with momentum p and the binding energy due tothe interaction via the Kerr nonlinearity. The most gen-eral fundamental soliton state is obtained from an arbi-trary superposition of these n-photon solitons accordingto Eq. (5) where the n-photon soliton itself can be in anarbitrary superposition of momentum eigenstates [6]:

f„(x„.. . , x„,t)= f dp g„(p)f„(x„.. . , x„)e

The function g„(p) is again a normalized square-integrable function. The dynamics of the soliton aredetermined by the energy spectrum. Note that themomentum distributions can be chosen independently ofn, yet Eq. (5) still describes only one soliton, as we willprove in Sec. III.

f if„(x„.. . , x„,t)i'dx; =1 .(6)

III. STABILITY OF SOLITONSAND THE HANBURY BROWN —TWISS EXPERIMENT

Substitution of (5) into the Schrodinger equation (4) re-sults in a Schrodinger equation for the n-particle wavefunction [5]:

n Q2+2c

BXJ L,J1~i(j&n

5(x —x;) f„. (7)

Thus, the quantum theory of pulse propagation in a fiberwith group-velocity dispersion and instantaneous self-phase modulation is equivalent to a gas of bosons in-teracting via a 5 potential. For bosons, the solutions f„have to be symmetric with respect to permutations of thecoordinates x;. If the constant c is negative, so that theinteraction between the bosons is attractive, Eq. (7) hasbound solutions. By using the Bethe ansatz method [6],one Ands that this Schrodinger equation has bound eigen-states. A subset of these bound states are characterized

The stability of the quantum soliton, composed of "ele-mentary bosons" can be proved by determining the signa-tures for a breakup of the soliton. A possible experimen-tal scheme to detect such a breakup of the quantum soli-ton is a Hanbury Brown —Twiss-type experiment, asshown in Fig. 1. If the soliton broke up into several dis-tinct groups of bosons, the hierarchy of possible correla-tion measurements, as indicated by Fig. 1, would have toshow a strong dependence on the normalized propagationtime t or fiber length l. The general solution of theQNSE, which is given by (5) and (11), gives an expressionfor the wave function of the field at the output of thefiber. In order to analyze the correlation functions onemeasures with the setup of Fig. 1, we express the fieldoperators at the detectors P, (x) in terms of the fieldoperator at the fiber output P(x) and the field operatorsP;"(x) at the empty vacuum ports of the beam splittersfor 0 + i ~ m. If we denote with r; and t; the transmission

48 QUANTUM-MECHANICAL STABILITY OF SOLITONS AND THE. . . 2363

t=o

Fiber

e(x)ly(E, x))

t=E

ep"(x) e","(x)l0& l0+

PD PD PD

A A A

Iii(x) Ii(x+x,) -. I~(x+'rm)

(x)=t, . tpg(L, x+r )

+t, t, roy~o"(x+r )

+ +t ir 2$" 2(x —(r +r 2)) .

Note that ~, to ~ denote the differences between thepropagation distances of the fields between the end of thefiber and the individual detectors (see Fig. 1). The corre-lation functions measured with the experimental setup ofFig. 1 for r, & r2 « . r„are given by [24]

A AtDetector current: l,-(x) = @,(x)

1P

(&i . -. &m l)

FIG. 1. A fundamental soliton is launched in a fiber and

propagates a distance l. When it leaves the fiber, a modified

Hanbury Brown —Twiss experiment is performed, which allowsus to measure photon-count correlation functions of arbitraryorder m. The case m =1 corresponds to the original HanburyBrown —Twiss experiment.

and reAection coefficients of the beam splitters and as-sume that the field between the beam splitter propagatesin free space, one derives for the relation between thefield operators (see Fig. 1):

Pp(x) =rod(x) tplkp (x)

Pi(x) =r, tpP(x +r, )+r, rpPp"(x +r, )

G(m +1)( r l)

= f (P(l), OiI(x)Ip(x+ri)

X ' ' I (x+r )ly(l), O&dx, (13)

with the intensity operators I;(x)=P;(x)P;(x).g(l) denotes the state of the field at the output of the fiberand 0 the vacuum state of the empty beam-splitter ports.The beam-splitter transformations (12) cause the fieldoperators at the individual detectors to commute witheach other, which is necessary since the fields at thedifferent output positions are independently measurableand have to commute. This allows normal ordering ofthe entire expression after substitution of relations (12)into (13), since each individual intensity is already normalordered. Then we can apply the field operators of theempty input ports onto the corresponding vacuum statesand obtain the time- and normal-ordered expression

=C t x x+~) x+~

X p(x +r ) . p( x+ r)p( )x~ 11(tt) &1 x,

(14)—t, P,"( )x,

I/2(x)=r2titplf(x +72)+r2tirplpp (x+T2)

+r2ti rifi"(x+(rz —ri ))—tzPz"(x), (12)

in terms of the field in the fiber. Note that C is a constantcontaining all the reAection and transmission coefficientsof the beam splitters. Since this constant is of no impor-tance, we set it to 1 for convenience.

For the general state (5), and using the symmetry of thewave functions f„,we obtain

G' +"(r . . . , r, l)= g ~a ~

' f d" 5(xr, —(x„,—x„)}i& ' ' '

& m& tf[( +1)]t.X . 5(r —(x„—x„))~f„(x„.. . , x„l)~ (15)

This equation has a simple interpretation. The square ofthe wave function gives the joint probability of findingthe n photons at the positions x;. In order to get a mea-

surement result different from zero, we have to have atleast I +1 photons in the field; if one of the m +1 pho-tons arrives at time x„, the other m photons have to ar-rive at x„+~; the counting of these events is provided

by the product of 5 distributions in (15). There aren!/[tt —(m +1)]!possible choices for choosing the m +1photons at the I +1 times out of a set of n photons.Equation (15) is valid for a general field of bosons. For

the case of a first-order soliton, the wave functions f„have the special form (11). The variable p in this solutionis the conjugate variable of the mean position of the n

photons since it is in the wave function in front of themean position coordinate:

=1y„=—gx) .j=1

The part of the wave function describing the bound statethen depends only on the n —1 relative coordinates:

2364 F. X. KARTNER AND H. A. HAUS 48

y =x —xn for 1 j~n —1. (17)

When the wave function (11) of the fundamental soliton state is introduced into (15), the integrand is expressed in termsof the mean position coordinate and relative coordinates. Therefore, after a change of variables [x;]~ [y; I with nochange of the integration volume, the integration over the mean position coordinate yn can be easily performed and re-sults in the integral

f dp dq [g„(p)]*g„(q)5(p—q)expIi [E(n,p) E(—n, q)]l j =1 . (18)

Note that the integration over the mean position coordinate has generated the 5-distribution in the momentum vari-ables and thus the dependence on time, i.e., dependence on propagation distance l, of the correlation function vanishes.Performing the additional integrations over the relative coordinates yn, - for 1 i m finally results in

p (~ —1)"lcl"Ig„l' exp c g Ir, I+

[n —(m +1)]!1)J

1&i &j&m

x f" dy& dy„~ +, ,exp cn —(m +1)

ly;I+i=1 l) J

1&i &j&n —(m+1)

m n —(m+1)ly,

—y;I+ & & lr, —y;I

A further simplification for general m is not immediately possible; however, it shows the independence of this wholehierarchy of photon-count correlation functions with respect to the propagation distance of the pulse I. This resultshows that the quantum-mechanical soliton described by the general state (5) shows stationary behavior with respect tothe complete hierarchy of photon-count correlation measurements possible by the generalized Hanbury Brown —Twissexperiment of Fig. 1. To establish a correspondence principle for these soliton states, we want to simplify the generalexpression (19) for the case of the original Hanbury Brown —Twiss experiment m = 1:

G' '(r, l)= g la„l'

exp(clrl)~ (n —1)! Icl"

z" (n —2)!

n 2

X f d, dy„zexp c g Iy, l+OO

1

n 2

ly,—y;I+ g lr —y;I

1)J1&i &j n —2

(20)

The integrals appearing in Eq. (20) are evaluated in the Appendix, and we obtain

G"'(r)= g l~. l'G"'(r),n ~2

(21)

with the intensity autocorrelation function of an n-photon soliton,

nIclexp — Ical

4

(n —2)/2+2 y exp[j'I«l ]

2 j—1

2

2

X ~—j (1+2j Ical ) 4j for n eve—n

2(22)

G(&)(r) —.n

2lclexp — Ical4

1 ' (n —1)/2exp[( j —

—,'

) Icr

I ]

2

2

—(j ——')'2 2

X ~—(j——,

')' [1+2(j——,')'I«l] —4(j ——,')' for n odd . (23)

QUANTUM-MECHANICAL STABILITY OF SOLITONS AND THE. . . 236S

For n =2, 3, and 4 we obtain the explicit expressions

G',"( r) = ~c~exp( —~cr~ ),G3 '(r) =4]c(exp( —2)cr()((ct)+ I),G+ '(r) = ]c(exp( —4[cr( )

(24)

the classical result at ~=0:3

G(z) (0)Ci, n (27)

In the quantum-mechanical case, we obtain from Eqs.(22) and (23),

X [16+3exp( jcrl)(12lcrl —2) j .qm, n (28)

Thus in the case of a very low photon number, the inten-sity autocorrelation function reproduces an exponentialdue to the exponential shape of the wave functions (8).

The classical fundamental soliton solution of the corre-sponding CNSE to (1) is given by [25]

Thus the difference between the normalized classical andquantum-mechanical intensity correlation functions isonly of order I /n, as can be seen from Fig 2.. This is incontrast to the integrals over the intensity autocorrela-tion functions

n c '" n'c'4(x, t)= exp i t ipot— fG,')'„(r)dr=n

f Gq' „(r)dr=n(n —1),

(29)

+LPO(X Xo)+leo

X sech (x —xo 2pot )—n[c/ (25)

where n is the equivalent photon number correspondingto the energy of the classical soliton, po the momentum,and xo and Oo the initial position and phase, respectively.Therefore, analyzing the intensity autocorrelation experi-ment with this classical solution would result in

G.", '. ( &r),= f+"dx&l@(x+r,t)l'lux, t)l'&

n'fc/4

nicer coth nlclr2

—2

no'Tsinh

(26)

CO

CV

~ U"CV

0.8

0.6

0.4

o.a

(&)Gcl, n 100

108642

Figure 2 shows the correlation functions of the quantumand classical solitons for different photon numbers. Thetime is normalized to the width of the correspondingclassical soliton with an equivalent photon number n, andthe correlation functions are normalized to the value of

which show a difference of order n as is necessary for ann-photon state. Thus the spatial distribution of the pho-tons is closer to a classical behavior. Figure 2 shows thatthe most striking difference between the classica1 andquantum-mechanical intensity autocorrelation functionsis the nondifferentiability of the quantum-mechanical re-sult at &=0. However, with increasing photon numberthe differences between the quantum and classical resultsdisappear rapidly. For n =10, the difference between thetwo functions has already dropped to l%%uo at v =0 accord-ing to Eqs. (27) and (28); for n =100 the classical andquantum correlation functions already agree at all timeswithin the resolution of Fig. 2.

Thus far, we have discussed only the case of an n-photon soliton. A superposition state of n-photon soli-tons with arbitrary momentum distributions g„(p) leadsto an intensity-correlation function that is just theweighted average of the individual intensity-correlationfunctions as shown by Eq. (21). Thus, if the probabilitydistribution of photon number peaks at n & 100 and van-ishes for low photon numbers, the intensity-correlationfunction approaches the classical result. This gives us acorrespondence principle for the quantum soliton, whichrelates it in a unique way to the classical soliton andproves that an arbitrary state of the form (5) with wavefunctions (11) shows solitary behavior with respect to theintensity autocorrelation function. Of course, this doesnot mean that all these states show classical behavior inthe limit of large photon numbers, as we will demonstratein Sec. IV.

J[ ~9,(P)~ = ~9(P)~

-2 0nick

2

FIG. 2. Normalized intensity autocorrelation functions for aclassical soliton and quantum solitons in a photon-number staten. FIG. 3. Momentum distribution of a Schrodinger-cat soliton.

F. X. KARTNER AND H. A. HAUS

IV. SCHRODINGER-CAT SOLITONSAND HOMODYNE DETECTION

Let us assume a soliton which has an arbitraryphoton-number distribution, but the momentum distribu-tion g„(p) for each individual photon-number state ischosen to be a double-peaked function such as

&,Adelay

A A.fiber

1

( 2$ )1/2 1/4( )= ex

+exp

Xexp( —inpxo),

(p —po)'

25p

(p+po)'26p

(30)

FIG. 5. Homodyne detection with a control pulse would al-low us to uncover a quantum superposition state according toEq. (31). The control pulse can be delayed by an amount 4poz,so that it can interfere with both parts of the superposition stateto achieve a nonvanishing detector current.

with po))5p (Fig. 3). With this momentum distribution,the general state (5) with wave functions (11) can be writ-ten as a superposition of two soliton states:

(31)

where ~1(t~ & denotes the state of a soliton moving withPomomentum po [6] and g & denotes the state of a soli-

Pp

ton moving with momentum —po, respectively. Offhand,one may suppose that the state (31) is the quantum repre-sentation of two classical first-order solitons of differentcarrier frequencies. This is not so, however. First, theHanbury Brown —Twiss experiment does not distinguishthis state from one with a momentum distribution withone single peak, whereas two classical solitons of differentcarrier frequencies would register differently in the exper-iment than one single soliton. Second, two solitons are,strictly speaking, manifestations of higher-order solitonsand these are not represented by (8) [6]. Since themomentum p determines the center frequency of the soli-ton, the individual states move with different group ve-locities due to dispersion. Thus, during propagation theinitial soliton state (31) which is a superposition state inmomentum will also split up into a superposition atdifferent positions which are a distance Az=4pol apart(Fig. 4). Since we allow for arbitrary photon statistics,the pulse can have a macroscopic meaning in the limit oflarge photon numbers and can therefore be considered aSchrodinger-cat state. Of course, the fiber loss woulddestroy such a macroscopic superposition state within avery short propagation distance [26].

In the intensity autocorrelation measurement in Sec.III, we always compared the soliton with itself; therefore,the momentum statistics did not enter the result. If wecould generate a soliton as in Eq. (31) and send a controlpulse parallel to the soliton (see Fig. 5) we could detectthis Schrodinger-cat soliton by using a homodyne detec-tor which measures the time delay between the controlpulse and the soliton.

V. CONCLUSION

We analyzed a generalized Hanbury Brown —Twiss ex-periment with solitons. It demonstrates clearly the sta-tionary properties of solitons with respect to photon-count correlation functions and therefore proves the sta-bility of an arbitrary fundamental soliton state. %"eshowed for the case of the intensity autocorrelation func-tion that, independent of the momentum statistics, aquantum soliton with a large photon number approachesthe correlation function given by the classical solitonsolution, which gives us a correspondence principle forsolitons. Whereas intensity autocorrelation measure-ments in the high-photon-number limit approach classi-cal behavior, the properties of a quantum soliton can de-viate strongly from the classical picture, as shown by theconstruction of a Schrodinger-cat soliton in Sec. IV. Thisdemonstrates that not all of the possible quantum solitonstates are useful for optical communication.

ACKNOWLEDGMENTS

x)Quantum superposition

This work was supported, in part, by ONR Grant No.N00014-92-J-1302 and Draper Laboratories Grant No.DL-H-441629. F.X.K. is on leave from the Lehrstuhl furHochfrequenztechnik, Technische Universitat Munchen;his support by the Alexander von Humboldt Stiftung isgratefully acknowledged.

APPENDIX

FIG. 4. During propagation, a soliton with a momentum dis-tribution according to Fig. 3 would split up into a quantum su-

perposition state describing a soliton at two different positionsin space.

In this appendix we evaluate the intensity autocorrela-tion function (20) for an arbitrary fundamental solitonstate. From Eq. (20) we obtain with the substitutionicky, =x, ,

QUANTUM-MECHANICAL STABILITY OF SOLITONS AND THE. . . 2367

G (r, t ) = g a„ I exp( —Ical )I(n —2, 0, Ical ),2 (n —1)t'

(n 2)(A1)

where the integral I is given by

n 2 n 2I(n —»a r)= f « . «.—exp —ia g lx;I — g lx;I+lr —x;I+

i=1 i=1 l, J1+i (j+n —2

lx, —x, l (A2)

Since this integral is invariant with respect to permutations of the variables x;, we can restrict the domain of integrationto —~ &x, ~x2 ~ . . xn 2 & ~ and account for the entire domain by multiplication with the number ofpermutations ( n —2 )! This domain can be further reduced into the subdomains —ao (x,

x 0 . xk ~~~xk+1+ . . . xn 2& ~, with O~m +k ~n —2, where we have assumed that ~+0.Then the absolute values in the exponents can be resolved within each domain and we get

n —2 k

I(n —2, a, r)=(n —2)! g g I, (m)I2(m, k)I3(k)exp[ r(2k —n+2—)],k=0m =0

(A3)

0 X3 X2 m

I&(m)= f dx f dxz . . f dx&exp —g (ia+2j n —1)x-j=l

k

I2(m, k)= f dxk . . f™~2dx+,exp — g (ia+2j n+1)x-.j=m+1

(A5)

I3(k)= f dx, +,f dx„+, f dx„,exp —g (ia+2j —n +l)x,1 Xa+ &

X 3 j=lThe integrals I, (m) and I3(k) can be found in the Appendixes of Refs. [15]or [6] and are given by

( 1)m 171

I, (m)=m! . , ia+(j n)—

1 n 21I3(k)=, n . . expI —[ia+(k+2)](n —k —

2)IVIII .(n —k —2)! . k+, ia+(j +1)

It can be proved by induction that the second integral is given by

k —m+1 k —m

I2(m, k) = g A (l, k, m)exp —g (ia+Zj +2m +n1)—l=1 j=l

with the coefficient

k —m J l —1 l —1

A(l, k, m)=( —1) +' n —g(ia+2r+2m n+1) n——g(ia+2r+2m —n+1)j=1 r=l j=1 r=j

(A6)

(A7)

(AS)

(A9)

Performing the sums in (A10) and putting everything into Eq. (A3), we obtainn —2 k k

( 1)k —m+II( n —2, a, r ) = ( n —2 )! g g ~ !(k —l )!( ——k )!(l — )!

l —m 1

+J —n .1

&a+J+I+m —n +11

k —m1

ia+j+1 .& +, ia+j +l+m n+2—

mxn, .j=ln 2

xj=k+1

XexpI —ia(n 2 l)r+[l (—l—+ l)(n —2)]—r] .

1)k

—m+I+1 k —m 1 1

(k —m —l+1)!(1—1)! . , ia+j +i+2m n+1 —. , ia+j+l+2m n—~

(A11)

Since the exponential depends only on the running index I, we change the summation indices so that the summationover I is the final summation with fixed boundaries:

2368 F. X. KARTNER AND H. A. HAUS 48

I(n —2, a, r) =(n —2)! g exp[ —ia(n —2 —l)r+ [I (—I +1)(n —2)]r]S,S2,1=0

with

(A12)

I( 1)m m

m!(I —m)!

n —I —2( 1)l+k

S2 = Y.k!(n —2 —I)!

Using the relation

I —m1 1

ia+j n—. , ia+j +I+m +1 n—2 k —m+1 1

k+I+1 i++j . I +1ia+j+l+m —n +2n(A13)

n —1 1 n —1 —m m

(j+ia) + (j —ia)=no m!(n —1 —m)!

from Ref. [15],we can perform the sums in Si and Sz and get

I 1S, =(I+1)+I =i j —(ia+I n+1)—

n —2 —I 1S2=(n —1 —I) Il j (ia—+ I + 1)

n —2 —II 1 1X( —1 —I) +

, j (i a+I —n+1—) 1, j —(ia+I+ 1)

Putting everything back into Eq. (A12), we getn —2

I(n —Z, a, r) =(n —2)! g (I +1)exp( ia(n —2 l)r+—[——[I +(I +1)(n —2)]r] )

1=0

(A14)

(A15)

(A16)

We can write the products in terms of Euler's gamma function and by using the mirror formula for the I function [27]

r(z)r(1 —z)= ."

sin(mz)

we obtainn —2 —I! 1 1

, =, j (ia+I —n+1) ~—=, j (ia+I+—1)(n —I —1 —ia)(n —2l —2 —ia)(l +1+ia)(—m)

~r( n —ia )

~sin( iam. )

(A17)

(A18)

Substituting this result into Eq. (A16) and changing the summation indices to I =(n —2)/2+ j for I ) (n —2)/2 andI =(n —2)/2 —j for I ((n —2)/2 in the case of an even number of photons n and to I =(n —1)/2+ j for I ) (n —1)/2and I = (n —1)/2 —j for I ((n —1)/2 for the case of n odd, we finally end up with

(n —2)! 7l 2exp —i a

~l (n —ia)~n

4—1

I(n —2, a, r)= .

nLA

2

(n —2)/2+ g exp[j r]

sin( i an. )

X 2 Re exp[ijar]2

2n j

m(2j+ia)j +lasin( iacr )

for n even

(A19)(n —2)! 11 2

exp —i a~r(n ia)~—

n

4—1

(n —1)/2exp[(j —

—,')'rlj=12

X 2 Re exp[i( j——,' )ar]

In the limit +~0, we obtain using L'Hospital's rule

n, , z rr(2J' —1+ia)—(j ,'+ ia——2 ' sin(iam )

for n odd .

(A20)

QUANTUM-MECHANICAL STABILITY OF SOLITONS AND THE. . . 2369

(n —2)!exp

(n —1)!

(n —2)!I(n —2, 0, r) = exp(n —1)!

n —14

T 4(n —2)/2

+2 g exp[ j~r]2 j=l

n —14

211

2

—j (1+2j ~) 4j —. for n even2

(A21)

(n —1)/2X2 g exp[(j —

—,') ~]i=~

2

r 2

—(j —

—,'

) [ 1+2(j ——,'

) r ]—4(j——,'

) ~ for n odd . (A22)

Substitution of this result into Eq. (A 1) leads to the intensity-correlation function for a general fundamental soliton.

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