Soliton squeezing and the continuum

618 J. Opt. Soc. Am. B/Vol. 17, No. 4 /April 2000 H. A. Haus and C. X. Yu

Soliton squeezing and the continuum

H. A. Haus and C. X. Yu

Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received May 17, 1999; revised manuscript received October 6, 1999

We review soliton perturbation theory with renormalized soliton operators. We analytically evaluate the ef-fects of the continuum on squeezing. Our results show that the contribution that is due to the continuumexhibits an oscillatory behavior and can be beneficial to squeezing. © 2000 Optical Society of America[S0740-3224(00)01304-7]

OCIS codes: 270.5530, 270.6570, 270.5290.

1. INTRODUCTIONSoliton squeezing has been extensively analyzedanalytically1–13 and demonstrated experimentally.13–17

Soliton squeezing was first numerically investigated bymeans of stochastic differential equations by Drummondand Carter.1 The stochastic approach introduced noisesources into the equations of propagation. Haus and Lai3

analyzed the problem of squeezed soliton vacuum genera-tion in a balanced Sagnac loop by using linearization ofthe nonlinear Schrodinger equation (NLSE). Becausethis equation is derivable from a Hamiltonian, it con-serves commutator brackets and thus does not requirethe introduction of noise sources.14,15 It was shown thatthe two formalisms lead to the same results.6 The ap-proach based on the linearized NLSE leads to linear dif-ferential equations for the operators. Because the solu-tions of linear operator equations do not involvecommutators, they are identical in form with solutions ofdifferential equations of classic c-number variables. Thiscorrespondence with classic evolution allows for simpleinterpretations of the quantum behavior.

The linearized equations were solved3 by expression ofthe solution as a superposition of the four soliton pertur-bations, i.e., photon number, phase, position, and momen-tum, and the continuum. Generally, squeezing is de-scribed by the evolution of the in-phase and quadraturecomponents of the electric field. These have the same di-mensions, unlike the operators of photon number andphase. To bring out more closely the correspondencewith the conventional approach to squeezing by means ofthe second-order nonlinearity, we renormalize the perti-nent operators. One welcome consequence of the renor-malization is that it establishes symmetry in the equa-tions of evolution between pairs of the four solitonperturbation parameters; in-phase and quadrature ampli-tudes; position and momentum.

Optimal detection of squeezing soliton vacuum requiresa local oscillator (LO) that is not a simple hyperbolic se-cant. Coupling to the continuum is thereby avoided. Inpractice it is much easier to use the secant hyperbolic ofthe squeezing pump for the LO. Under these conditions,coupling to continuum is unavoidable. The question thenarises as to the effect of this coupling, which has been par-

0740-3224/2000/040618-11$15.00 ©

tially addressed in Refs. 11 and 16. The coupling to thecontinuum leads to an oscillatory dependence of shot-noise reduction on the Kerr-induced phase shift. In fact,the coupling to the continuum can provide a (small) im-provement in shot-noise reduction by virtue of the factthat the noise in the continuum is correlated with thesoliton fluctuations. The research on amplitudesqueezing17–19 has also dealt with the contribution of thecontinuum.20,21 Oscillatory behavior has been found inthe amplitude squeezing experiments in an asymmetricSagnac loop.20,21 In our case the variation is due to twobeating effects: the self-beating of the continuum andthe beating between the continuum and the soliton.

In Section 2 we review soliton perturbation theory ofRef. 3. In Section 3 we renormalize the operators to castsoliton squeezing in the linearized approximation into astandard Bogolyubov transformation. In Section 4 we re-view the orthogonality properties of the continuum. InSection 5 we analyze the detection of a squeezed solitonwith the choice of local oscillator that rejects the con-tinuum in preparation to the main thrust of this paper,namely, the detection of the squeezed soliton with a hy-perbolic secant LO pulse as is usually done in practice, be-cause then the LO is directly available from the squeezingpump. This pulse couples to the continuum, which mayadd to, or subtract from, the noise level.

2. QUANTIZED NONLINEARSCHRODINGER EQUATION AND ITSLINEARIZATIONThe nonnormalized Heisenberg equation of motion is

]

]ta~x ! 5

i

2

d2v

db2

]2

]x2 a~x ! 1 iKa†~x !a~x !a~x !. (2.1)

This is the quantized NLSE. a(x) is an annihilation op-erator that annihilates a photon at position x; a†(x) is acreation operator that creates a photon at position x.The classical analogs of a(x) and a†(x) are the complexamplitude of the pulse envelope and its complex conju-gate; d2v/db2 is the second derivative with respect to thepropagation constant v(b); K is the Kerr parameter.

2000 Optical Society of America

H. A. Haus and C. X. Yu Vol. 17, No. 4 /April 2000 /J. Opt. Soc. Am. B 619

The quantized form of the NLSE was solved rigorouslyby use of the Bethe ansatz.2 An approach that leads tosimple analytic expressions and that permits physical in-sight is based on the linearization approximation.3 Onesets for the operator a(x)

a~x ! 5 a0~x ! 1 Da~x !, (2.2)

where the first term is a c number and the second is anoperator that takes over the commutation relation ofa(x). Thus

@Da~x !, Da†~x8!# 5 d ~x 2 x8!. (2.3)

Replacement (2.2) is rigorous and by itself does not implyany approximation. Approximations are made whena0(x) is made to obey the NLSE and the equation is lin-earized in terms of Da(x). Thus a0(x) obeys the equa-tion

2i]

]ta0 5

C

2

]2

]x2 a0 1 Ka0* a0a0 , C 5d2v

db2 . (2.4)

The solution is

a0~t, x ! 5 A0 expF iS KA02

2t 2

C

2p0

2t 1 p0x 1 u0D G3 sechS x 2 x0 2 Cp0t

jD , (2.5)

with the constraint that

A02j2 5 C/K. (2.6)

The solution has four arbitrary integration constants, A0 ,p0 , u0 , and x0 . These have been chosen on account oftheir interpretation as average amplitude, momentum,phase, and position.

The average photon number n0 is given by* dxa0* (x)a0(x):

E dxua0~t, x !u2 5 E uA0u2 sech2S x 2 x0 2 Cp0t

jD dx

5 2uA0u2j 5 n0 . (2.7)

In the subsequent analysis we shall set p0 5 u0 5 x05 0, which simply means that we have chosen a coordi-nate system whose origin is at the pulse center, we haveset the phase equal to 0, and we have picked a momentum(or carrier frequency) that coincides with the nominal car-rier frequency v0 .

When ansatz (2.2) is introduced into the NLSE andterms of order higher than first in Da and Da† aredropped, one obtains a linear equation of motion for thesetwo operators:

2i]

]tDa 5

C

2

]2

]x2 Da 1 2Kua0u2Da 1 Ka02Da†. (2.8)

The equation couples Da and Da† in a way that is char-acteristic of a parametric process. Linear equations ofmotion of an operator are in one-to-one correspondencewith linear equations of motion of the classic evolutionequation. In the integration of such equations one doesnot encounter products of operators, for the inclusion of

which one would have to use the commutation relations.Hence the integration can proceed classically, as if the op-erators were c numbers.

We note that Da consists of two parts: a part Dasolthat describes the change of the soliton parameters, i.e., apart that is associated with the soliton, and a part Dacontthat is not associated with the soliton, the continuumpart:

Da 5 Dasol 1 Dacont . (2.9)

The soliton perturbation is with respect to the four de-grees of freedom of the soliton: the photon number, thephase, the momentum, and the position. These pertur-bations are all operators. They are functions of x. A so-lution of Eq. (2.8) has been obtained2 through separationof variables by use of the solutions of the classic form ofthe NLSE as a guide. The perturbation is written as asuperposition of operators with associated functions of x.The operators for photon number and phase, Dn and Du,have the usual interpretation. The operator of the posi-tion, D x, is associated with the displaced position x0 ; theoperator for momentum, with the shift from carrier fre-quency p0 . A carrier frequency shift Dp corresponds to achange of propagation constant Db, where \b is the mo-mentum. It is important to note that the change of mo-mentum of a wave packet with the average number ofphotons n0 is equal to n0Dp. Hence it is natural to writethe perturbation in the form

Dasol 5 @Dn~t !fn~x ! 1 Du~t !fu~x ! 1 D x~t !fx~x !

1 n0Dp~t !fp~x !#expS iKuA0u2

2t D . (2.10)

The functions fi(x) are the derivatives of the solitonevaluated at t 5 0:

fn~x ! 51

2A0jF1 2

x

jtanh~x/j!Gsech~x/j!, (2.11)

fu~x ! 5 iA0 sech~x/j!, (2.12)

fx~x ! 5A0

jtanh~x/j!sech~x/j!, (2.13)

fp~x ! 5 i1

2A0jx sech~x/j!, (2.14)

where we have fixed the phase by defining A0 real andpositive. When ansatz (2.10) is introduced into the lin-earized NLSE one finds that no new functions are gener-ated by the derivatives with respect to x. Equating thecoefficients of the functions fQ(x), Q 5 n, u, x, p, onefinds equations of motion for the operator soliton pertur-bations:

d

dtDn 5 0, (2.15)

d

dtDu 5

1

2K

Dn

j, (2.16)


d

dt

D x

j5 2

C

j2 Dp, (2.17)

d

dtDp 5 0. (2.18)

This is a review of the approach presented in Ref. 2.Squeezing by means of a second-order nonlinearity oper-ates on the uncertainty ellipse of the in-phase andquadrature components of the electric field. Vacuumfluctuations, a stationary process, are represented by anuncertainty circle. Squeezing is observed when the circleis deformed into an ellipse of the same area. This pro-cess is described by a Bogolyubov transformation. Torepresent soliton squeezing by a third-order nonlinearityin terms of a Bogolyubov transformation it is necessary torenormalize the photon number and phase operators intoin-phase and quadrature component operators. There isan analogous squeezing process that operates on the un-certainty ellipse of momentum and position. It is equallyconvenient to express this process as a Bogolybov trans-formation and thus introduce renormalized momentumand position that have the same dimension.

3. RENORMALIZATION OF THE SOLITONOPERATORSPerturbation operator Da(x) has the commutator@Da(x), Da†(x8)# 5 d (x 2 x8) and thus has dimensionsof inverse length to the half-power. Photon-number per-turbation Dn is given by Dn 5 D(2A0

2j) 5 4A0DA0j1 2A0

2Dj 5 2A0DA0j, where we have used the areatheorem to relate the pulse-width change to the pulse-amplitude change. Next, consider a continuous wave ofamplitude A0 and its associated photon number n05 A0

2 . The change in photon number is Dn5 2A0DA0 . When it is quantized, the perturbation DA0

will be replaced by the in-phase operator, DA0 → DA1 .This fact and the dimensions of the Da(x) operator sug-gest that the soliton perturbation DA0Aj is to be replacedby

DA0Aj → DA1 . (3.1)

Its associated expansion function is changed by the renor-malization from the expansion function of the photonnumber perturbation [Eq. (2.11)]:

f1~x ! 51

AjF1 2

x

jtanhS x

jD GsechS x

jD . (3.2)

The same approach suggests the definition of the quadra-ture component as

A0DuAj → DA2 . (3.3)

We find for the expansion function

f2~x ! 5i

AjsechS x

jD . (3.4)

A similar renormalization is possible for the perturba-tion operators of position and momentum. As we shall

see, it is convenient to change the commutation relationby a factor of 1/2. This is accomplished by the identifica-tion of the new operators DX 5 A0D x/Aj and DP5 n0DpAj/2A0 . The commutator is now

@DX, DP# 5 i/2. (3.5)

The respective perturbation functions become

fX~x ! 51

AjtanhS x

jD sechS x

jD , (3.6)

fP~x ! 5i

Aj

x

jsechS x

jD . (3.7)

Expansion (2.10) of the pulse is now in the form

Dasol 5 @DA1~t !f1~x ! 1 DA2~t !f2~x ! 1 DX~t !fX~t !

1 DP~t !fP~x !#expS iKuA0u2

2t D . (3.8)

The adjoint functions are defined by

ReF E dxfIm* ~x !fn~x !G 5 dmn (3.9)

for m, n 5 1, 2, P, Q. It is easy to show that the equa-tion adjoint to Eq. (2.8) is

i]

]tDa 5

C

2

]2

]x2 Da 1 2Kua0u2Da 2 Ka02Da†.

(3.10)

Note the change of sign in the last term. There is asimple physical reason for the form of the adjoint: Thephysical process is parametric pumping. Such pumpingcan produce growing and decaying excitations. Energy isnot conserved in growth or decay. However, cross energycan be conserved between a growing and a decaying solu-tion. The growing and decaying solutions are shifted inphase with respect to the pump. This shift is the originof the phase change between the equation and its adjoint.

The solutions of the adjoint equation, properly normal-ized, are

fI1~x ! 51

AjsechS x

jD , (3.11)

fI2~x ! 5i

AjF1 2

x

jtanhS x

jD GsechS x

jD , (3.12)

fIX~x ! 51

Aj

x

jsechS x

jD , (3.13)

fIP~x ! 5i

AjtanhS x

jD sechS x

jD . (3.14)

The commutator of the in-phase and quadrature com-ponents is


@DA1 , DA2# 5 iE dxfI1* ~x !E dx8fI2* ~x8!

3 @Da ~1 !~x !, Da ~2 !~x8!#

5 21/2 E dxfI1* ~x !E dx8fI2* ~x8!d ~x 2 x8!

5 i/2 (3.15)

and has the expected value.It is clear from the preceding discussions that the ex-

pansion of a pulse excitation into a soliton part and a con-tinuum part is an expansion in a complete set of modes.These modes are phase dependent; the components inphase with the pulse a0(t, x) are different from those inquadrature. They form a complete set into which any ex-citation can be decomposed and whose amplitudes arequantized. Of course, the decomposition makes physicalsense only when the expansion represents perturbationsof a hyperbolic secant pulse. But the pulse need not be asoliton; e.g. it could be the hyperbolic secant pulse pro-duced in the output of a beam splitter with a soliton im-pinging upon one of its input ports.

It is of interest to determine the mean-square fluctua-tions of the soliton perturbation parameters if the back-ground is zero-point fluctuations. In this way one findsthat

û~DA1!2u& 5 E dxfI1* ~x !E dx8fI1~x8!@Da ~1 !~x !Da ~1 !~x8!#

5 1/4 E dxfI1* ~x !E dx8fI1~x8!d ~x 2 x8!

5 1/4 E dxufI1* ~x !u2 5 1/2. (3.16)

The mean-square fluctuations are twice the minimum-uncertainty value for equal in-phase and quadrature fluc-tuations. The remaining three fluctuations can be com-puted analogously. It is clear that they involve thevalues of the integrals

E dxufIQ~x !u2 5 2, ~1.214!, p2/6, 2/3; Q 5 1, 2, X, P.

(3.17)

The uncertainty products are

û~DA1!2u&û~DA2!2u& 5 2.43/16, (3.18)

ûDX2u&ûDP2u& 5 1.09/16. (3.19)

The in-phase and quadrature fluctuations are uncorre-lated:

ûDA1DA2 1 DA2DA1u& 5 0. (3.20)

To appreciate better the significance of the in-phasefluctuations, we return to relation (3.1) and take note ofthe fact that the photon-number fluctuations are given by

Dn 5 2A0DA0j → 2A0AjDA1 . (3.21)

Thus the mean-square photon-number fluctuations are

^Dn2& 5 2A02j2^DA1

2& 5 ^n&. (3.22)

They have the Poisson value. Hence the in-phase fluc-tuations of a soliton with twice the minimum value are, infact, the fluctuations associated with a Poisson distribu-tion of photons.

The renormalization has changed the uncertainty el-lipse. In the photon-number phase description, thephoton-number fluctuations were at the Poisson value;the phase fluctuations were larger than the minimumuncertainty.3 In the in-phase and quadrature descrip-tion the amplitude fluctuations are excessive, whereas thequadrature fluctuations are close to the minimum value.This shows that the description of squeezing is dependenton the representation. In fact, the minimum-uncertaintyellipse of the momentum and position of the particle isplotted along axes of different dimensions, and thus theshape of the ellipse is not an indication of squeezing. Itis only when the noncommuting variables are of the samedimensions and of the same character, such as the in-phase and quadrature components of the electric field,that squeezing can be identified.

The stationary character of zero-point fluctuationsguarantees equal in-phase and quadrature mean-squarefluctuations of each mode of magnitude 1/4, each givingan uncertainty circle of radius 1/2 and an area p/4 (defin-ing the circle as the locus of points of probability e21/2).The projection of zero-point fluctuations into the solitonconverts the circle into an ellipse of area greater than p/4.The fluctuations along the major and minor axes of the el-lipse are uncorrelated.

The position and momentum operators obey not thestandard commutator relation but a new one, in one-to-one correspondence with the commutator of the in-phaseand quadrature components. The renormalization leadsto new equations of motion for the operators. The deri-vation is parallel to that of Eqs. (2.15)–(2.18) and leads tothe same form of the equations but with new coefficients:

d

dtDA1 5 0, (3.23)

d

dtDA2 5

C

j2 DA1 , (3.24)

d

dtDX 5

C

j2 DP, (3.25)

d

dtDP 5 0, (3.26)

where we have used the area theorem KA02 5 C/j2.

The initial conditions for the equations of motion for theoperators are evaluated by projection of the excitationsDa(x) and Da†(x) at t 5 0. The renormalization has ledto a welcome symmetrization between the pairs of equa-tions that couple in-phase to quadrature amplitudes andrenormalized position to momentum observables.

4. THE CONTINUUMThe four soliton perturbation operators describe only partof the evolution of the field; the continuum describes therest. Gordon22 derived the orthogonal functions in terms


of which the continuum can be expressed. They obey Eq.(2.8). Outside the time slot occupied by the soliton theyare of the simple form exp(2iVx) and have the same am-plitude on both sides of the time slot. In the interval ofoverlap they change their amplitude and phase:

fc,s 5 c@V2 2 2iV tanh~x ! 2 tanh2~x !#exp 2 i~Vx

1 V2t/2! 1 c* sech2~x !exp~it !exp iS Vx 1V2

2t D ,

(4.1)

where

c 5 1 for the in-phase component,

c 5 i for the quadrature component.

We have set C 5 K 5 1, as we shall do henceforth tokeep the notation simple. There are two types of excita-tion:

(a) In phase with the soliton, denoted by the sub-script c (reminiscent of a cosine). These excitationschange the depth of the well inasmuch as they affect thenet intensity.

(b) In quadrature to the soliton, denoted by sub-script s (reminiscent of a sine). These excitations do notchange the depth of the well and are the well-known so-lutions of linear scattering from a hyperbolic secantwell.23

The adjoint functions satisfy adjoint equation (3.10).They are

fIc,s 5 c@V2 2 2iV tanh~x ! 2 tanh2~x !#exp 2 i~Vx

1 V2t/2! 2 c* sech2~x !exp~it !exp iS Vx 1V2

2t D ,

(4.2)

where

cI 51

~1 1 V2!for the in-phase component,

cI 5i

~1 1 V2!for the quadrature component.

The continuum functions are orthogonal to the solitonperturbation functions in the sense that

ReF E dxfQ~x ! fIc,s* ~x, t 5 0 !G 5 0, Q 5 1, 2, X, P.

(4.3)

The functions fc(V, x, t) are orthogonal to fIs(V, x, t), andthey obey the orthonormality condition

ReF E dxfIc* ~V, x, t !fc~V8, x, t !G5 ReF E dxfIs* ~V, x, t !fs~V8, x, t !G5 2pd ~V 2 V8!. (4.4)

The soliton perturbation functions fQ(x), Q 5 1, 2, X, Pand fc,s(V, x, t 5 0) form a complete set of functions interms of which any excitation can be expanded. Thecross orthogonality with their adjoints permits evaluationof the operator coefficients for any given initial conditions.In this way one can express the continuum excitation att, x:

Dacont 5 E2`

` dV

2p@Fc~V!fc~x, V, t ! 1 Fs~V!fs~x, V, t !#,

(4.5)

where the operators Fc,s(V) are obtained by projectionfrom the initial condition Da(x, t 5 0) and Da†(x, t5 0):

Fc,s~V! 51

2E

2`

`

dx@Da~x !fIc,s* ~x, V, t !

1 Da†~x !fIc,s~x, V, t !#. (4.6)

An initial Gaussian pulse would have a continuum con-tribution. In the case of soliton propagation, the evolu-tion of the basis functions is simple. The Gaussian pulsewould evolve into a hyperbolic secant pulse; the remain-der would radiate away as continuum. If the propaga-tion is not along a guide characterized by the NLSE, theevolution of this basis set would be complicated and use ofthe expansion may not be useful. However, for the pas-sage of the pulse and the continuum through a beamsplitter the description in terms of the orthonormal set ofmodes is convenient.

5. SOLITON SQUEEZING IN A FIBERPropagation of a soliton along a lossless dispersive fiberleads to squeezing of the soliton fluctuations. Squeezingoccurs as a result of the coupling between operators. Aswe pointed out in Section 3, the uncertainty locus of thein-phase and quadrature components of a soliton in azero-point fluctuation background is an ellipse, not acircle. The major and minor axes of the ellipse arealigned with the in-phase and quadrature axes, and thein-phase and quadrature fluctuations are uncorrelated.This initial ellipse is distorted by the squeezing processowing to the coupling of the in-phase component to thequadrature component. Within the linearization ap-proximation, the ellipse is stretched in the direction of thequadrature component, as shown in Fig. 1. Its area re-mains constant. The in-phase and quadrature fluctua-tions become correlated. The fluctuations along the ma-jor and minor axes are uncorrelated. This we nowproceed to show.

The solutions of Eqs. (3.23) and (3.24) are


DA1~t ! 5 DA1~0 !, (5.1)

DA2~t ! 5 DA2~0 ! 1 2F~t !DA1~0 !, (5.2)

where F(t) [ 1/2KA02t 5 1/2(C/j2)t is the classic soliton

phase shift.Equations (5.1) and (5.2) describe the evolution of the

uncertainty ellipse in the plane of the in-phase andquadrature components. The mean-square deviationsalong the in-phase and quadrature directions are, respec-tively,

^@DA1~t !#2& 5 ^@DA1~0 !#2&, (5.3)

^@DA2~t !#2& 5 ^@DA2~0 !#2& 1 4F2^@DA1~0 !#2&, (5.4)

with the cross correlation

1/2^DA1~0 !DA2~t ! 1 DA2~0 !DA1~t !& 5 2F^DA12~0 !&.

(5.5)We can treat the process of squeezing in a formal way

by establishing the correspondence of solutions (5.1) and(5.2) with the Bogolyubov transformation. We have

DA~t ! [ DA1~t ! 1 iDA2~t ! 5 m~t !DA~0 ! 1 n~t !DA†~0 !,

(5.6)with

m 5 1 1 i2F~t !, n 5 i2F~t !. (5.7)

Perturbation (5.6) accompanies the soliton pulse a0(t, x).We are now ready to analyze the generation of

squeezed soliton vacuum by the setup illustrated sche-matically in Fig. 2.24,25 The transform-limited hyperbolicsecant pulse is incident upon one of the ports of the Sag-nac loop acting as a nonlinear Mach–Zehnder interferom-eter. A Sagnac loop is chosen so index fluctuations of thefiber, which are slow compared with the transit time, donot lead to imbalance of the interferometer. The singlecoupler doubles as both an input and an output coupler.The collision of the two pulses at the symmetry point hasa negligible effect because the Kerr effect is weak andnonlinear phase shifts are accumulated only over fibersthat are many meters long. The energy of the hyperbolicsecant pulse is adjusted such that the two pulses in thetwo arms propagate as solitons. The zero-point fluctua-

Fig. 1. Deformation of the uncertainty ellipse.

tions that accompany each of the two pulses are producedby the superposition of the vacuum incident from port (b)and the vacuum accompanying the pulse in input port (a).The fluctuations that accompany each pulse are incoher-ent with each other. (The situation is analogous to thesplitting of thermal power by a beam splitter, each port ofwhich has equal thermal excitations.) The fluctuationsare squeezed as indicated by Eq. (5.6).

On their return to the coupler, the classic (c-number)excitation exits into port (a) and the squeezed zero-pointfluctuations are superimposed incoherently into port (b).The classic excitation is used as the LO in a balanced de-tector after an (optional) reshaping in the pulsetransformer.26 The noise that accompanies the LO pulsecancels. We assume that the reshaping produces the LOwaveform:

ifL~x ! 5 1/2@cos c fI1~x ! 1 sin c fI2~x !#exp~ic!, (5.8)

which can be put into the ideal form for the purpose ofprojecting out a linear combination of DA1(t) and DA2(t).In a real experiment the LO will of course produce gain G.For convenience Eq. (5.8) is normalized for G 5 1. Thesqueezed vacuum fluctuations emerging from output port(b) are projected out in the balanced detector, resulting inthe net charge operator25

DQ

q5 i E dx@fL* ~x !D asol 2 fL~x !D asol

†#

5 @cos cDA1~t ! 1 sin cDA2~t !#

5 1/2@exp~2ic!DA~t ! 1 exp~ic!DA†~t !#

5 1/2$exp~2ic!@mDA~0 ! 1 nDA†~0 !#

1 exp~ic!@m* DA†~0 ! 1 n* DA~0 !#%, (5.9)

where q is the electrical charge. The mean-square fluc-tuations of the detector current follow from Eq. (5.9).Equation (5.9) expresses the normalized difference chargein two ways: (i) as the projection of a vector with compo-nents DA1(t) and DA2(t) onto an axis inclined at an anglec with respect to the 1 axis and (ii) as the sum of thephase-shifted squeezed input excitations mDA(0)1 nDA†(0). The two representations are equivalent,but in particular applications one may be more conve-nient than the other. We shall determine the degree ofsqueezing and antisqueezing from representation (i).The mean-square fluctuations of the charge are

Fig. 2. Squeezing setup; (a), (b), input and output ports, respec-tively.


ûDQ2~t !u&

q2 5 cos2 cûDA12~t !u& 1 sin2 cûDA2

2~t !u&

1 sin~2c!1/2ûDA1~t !DA2~t !

1 DA2~t !DA1~t !u&. (5.10)

If the projection @ûDQ2(t)u&#1/2/q is plotted in the (1)–(2)plane as a function of orientation angle c, an ellipse istraced out, the locus of the root-mean-square deviation ofthe Gaussian distribution of ûDQ2(t)u&/q2. According toEq. (5.2) the component in direction 1 remains un-changed, whereas the component in direction 2 shifts pro-portionally to @ûDA1

2(0)u&#1/2. The mean-square fluctua-tions along the two axes are

ûDA12~0 !u&, ûDA2

2~0 !u& 1 ûDA12~0 !u&4F2. (5.11)

The cross correlation is

1/2ûDA1DA2 1 DA2DA1u& 5 2FûDA12u&. (5.12)

The probability distribution of the normalized differencecharge in the (1)–(2) plane with coordinates j1 and j2 is aGaussian, given by

p~j1 , j2 , t ! } exp 21

2 F j12

s11~t !1

j22

s22~t !1

2j1j2

s12~t !G ,

(5.13)where

The product of the eigenvalues is

l1l2 5 hûDA12~0 !u&2 (5.17)

and is constant, independent of the degree of squeezing.The squeezing and antisqueezing are illustrated in Fig. 3.

Fig. 3. Squeezing and antisqueezing (the minor and major axesof the squeezing ellipse) as functions of 2F.

Fig. 4. Root-mean-square fluctuations as a function of the phaseangle with respect to the LO: 2F 5 2, 4, 8.

Fs11~t ! s12~t !

s21~t ! s22~t !G 5 F ûDA1

2~t !u& 1/2ûDA1~t !DA2~t ! 1 DA2~t !DA1~t !u&

1/2ûDA1~t !DA2~t ! 1 DA2~t !DA1~t !u& ûDA22~t !u&

G5 ûDA1

2~0 !u&F 1 2F~t !

2F~t ! h 1 4F2~t !G (5.14)

and h 5 ûDA22(0)u&/ûDA1

2(0)u& 5 0.607. The Fouriertransform of the probability distribution expressed ink-space, the characteristic function, is of the form

C~k1 , k2 , t ! } exp 21/2@ s11~t !k12 1 s22~t !k2

2

1 2s12~t !k1k2#. (5.15)

The quadratic form in the exponent of the characteristicfunction can be diagonalized by a reorientation of theaxes. A coordinate transformation into new orthogonalcoordinates k18 and k28 finds the mutually orthogonal di-rections along which the fluctuations are uncorrelated.These are the major and minor axes of an ellipse. Thetransformation is a unitary transformation of the matrix,which leaves the eigenvalues of matrix (5.14) invariant.The eigenvalues are

l6 5 ûDA12~0 !u&H 1 1 h 1 4F2

2

6 F S 1 1 h 1 4F2

2 D 2

2 hG1/2J . (5.16)

With zero phase shift, the fluctuations in the 1 directionare shot-noise fluctuations. These are equal to twice thezero-point fluctuations of 1/4. In the orthogonal directionthe fluctuations are less, but they are still larger than 1/4.As the nonlinear phase shift increases, the branch thatrepresents shot noise at F 5 0 shows monotonically in-creasing fluctuations, whereas the orthogonal directiondecreases and reaches zero asymptotically. Figure 4shows the fluctuations as a function of phase angle c fordifferent degrees of squeezing and antisqueezing. Thisfigure shows that the regime of phase angle within whicha large degree of squeezing is observed becomes increas-ingly narrower as the degree of squeezing is increased.The greater the degree of squeezing, the harder it is tofind the squeezing angle and stabilize the system at thatangle.

6. CONTINUUM CONTRIBUTION TOSQUEEZINGThe projection with a LO orthogonal to the continuum27

permitted us to evaluate the perturbation of the soliton


without consideration of the continuum. In practice it isinconvenient to reshape the pulse for use in the LO. Anyreshaping introduces phase delays that may undergo fluc-tuations and prevent optimization of the relative phasebetween the LO and the squeezed vacuum. Use of thehyperbolic secant pulse as the LO pulse leads to detectionof the continuum contribution. We now show that thepenalty incurred when a hyperbolic secant LO pulse isused is not severe and, in fact, that at some phase shiftsthe noise is less than for the case of LO pulse that is or-thogonal to the continuum.

The first-order perturbation of the pulse is composed ofthe soliton perturbation and the continuum:

Da 5 Dasol 1 Dacont . (6.1)

When the LO pulse is a hyperbolic secant,

ifL~x, t ! 5 ~1/2!sechS x

jD exp~it/2!exp~ic!

5 ~1/2!fI1~x, t !exp~ic!. (6.2)

The function fI1(x, t) is the adjoint projection that evalu-ates DA1(t) from the perturbation Da(x, t). In addition,the LO waveform contains the phase factor exp(ic). Wefirst look at the contribution to the difference current ofthe soliton part of the squeezed radiation. Relation(5.15) must now be modified to take into account that thequadrature component is projected out with this particu-lar function, namely, sin c fI1(x, t), rather than sin c fI2(x, t).We find that Eq. (5.9) changes to

DQsol

q5 i E dx@ fL* ~x, t !Dasol 2 fL~x, t !Dasol

†#

5 @cos cDA1~t ! 1 2 sin cDA2~t !#. (6.3)

The mean-square fluctuations that are due to the solitonpart are

ûDQ2~t !u&sol

q2 5 cos2 cûDA12~t !u& 1 4 sin2 cûDA2

2~t !u&

1 sin~2c!ûDA1~t !DA2~t !

1 DA2~t !DA1~t !u&, (6.4)

with the mean-square fluctuations of the in-phase andquadrature components at t 5 0 the same as before.The matrix s ij

(sol, sol) analogous to Eq. (5.14) is

Fs11~t ! s12~t !

s21~t ! s22~t !G sol,sol

5 ûDA12~0 !u&F 1 4F~t !

4F~t ! 4@h 1 4F2~t !#G , (6.5)

where h 5 0.607. It is of interest to determine the majorand minor axes of the squeezing ellipse that are due tothe soliton noise alone, as affected by the nonideal LOwaveform. They are shown in Fig. 5. As the squeezingincreases, the minor axis of the ellipse turns increasinglymore closely into the (1) direction. In this direction, thesech LO is the projection function orthogonal to the con-tinuum. For this reason we see that the squeezing in-

creasingly approaches the values of Eq. (5.16) with in-creasing phase shift F. However, an undesirableconsequence of the increased coupling to DA2(t) is thenarrowing of the range of angles over which squeezing isobserved, as shown in Fig. 6, where we compare thesqueezing as a function of angle as evaluated from Eqs.(6.5) and (5.14) for F 5 1. This narrowing is due to thesignificant increase in the value of the antisqueezing ei-genvalue.

The hyperbolic secant LO waveform couples to the con-tinuum as well. We encounter a new ingredient of thedetection process: Beat terms of the continuum with thesoliton noise are found. This is the consequence of thefact that the noise within the soliton does not commutewith the noise of the continuum:

@Dasol , Dacont†# 1 @Dacont , Dasol

†# Þ 0. (6.6)

The partial coherence between the noise in the con-tinuum and the soliton noise is the consequence of thefact that the linearized NLSE is not self-adjoint. We con-struct a matrix of the self-correlations and cross correla-tions of the terms that multiply cos c and sin c, respec-tively, as in Eq. (5.14). To systematize this step weconstruct separately s ij

(sol, sol) , s ij(cont, cont) , and

s ij(sol, cont) , three matrices, from the sum of which we may

construct the major and minor axes of the squeezing el-lipse. Matrix s ij

(sol, sol) was given in Eq. (5.14). Notethat the only nonzero components of the matricess ij

(cont, cont) and s ij(sol, cont) are the 22 components, by vir-

tue of the fact that DQcont makes no contribution to thecos c term. The evaluation of s 22

(cont, cont) and s 22(cont, sol)

is carried out in Appendix A. The results are

s 22~cont, cont! 5

p

8E dV sech2S p

2V D cos2@~1 1 V2!t/2#,

(6.7)

Fig. 5. Minimum and maximum fluctuations of the soliton aloneas detected by a LO of hyperbolic secant shape. Comparisonwith ideal LO use.

Fig. 6. Fluctuations of the soliton alone as detected by a LO ofhyperbolic secant shape as a function of phase angle with respectto the LO, F 5 1.


s 22~sol, cont! 5

1

4E dV

1

~1 1 V2!F2

3p 1

p2

6V tanhS p

2V D

1p2V3

6tanhS p

2V D Gsech2S p

2V D

3 cos@~1 1 V2!t/2#. (6.8)

s22(cont, cont) and s 22

(cont, sol) are plotted in Figs. 7(a) and7(b), respectively. Both quantities oscillate, dependingon the phase between the LO and the continuum.

As the phase F increases, this contribution getssmaller and smaller. Collecting the three matrices, wemay evaluate the eigenvalues of the sum matrix that givethe squares of the major and minor axes of the squeezingellipse. Figure 8 shows the amount of squeezing in thecase of a LO pulse orthogonal to the continuum, without

Fig. 7. Matrix elements (a) s 22(cont, cont) and (b) s 22

(sol, cont) asfunctions of phase F. Note the beats.

Fig. 8. Minimum fluctuations detected by a LO that is orthogo-nal to the continuum and by a hyperbolic secant LO with andwithout a continuum.

the continuum and a sech LO, and with the contributionof the continuum. We find that the continuum improvesor reduces the amount of squeezing relative to both of thefirst two cases, depending on the phase between the con-tinuum and the soliton. However, as before, we findthat, at large degrees of squeezing, all three curves ap-proach one another. In this limit, inclusion of the con-tinuum has a negligible effect on the antisqueezingbranch, which is already a large number.

7. SUMMARYIn this paper we have renormalized the four soliton per-turbation parameters in such a way that, pairwise, theequations for the noncommuting observables becameidentical in form. The dimensionless expansion func-tions permit the quantization of any hyperbolic secantpulse in terms of a complete set of expansion functions,including a pulse that is not a soliton, such as the pulseemitted into a linear medium from the end of a fiber. Ofcourse, the evolution equations of the operator coefficientsmust be changed to accommodate the change of the me-dium of propagation.

The renormalization cast the squeezing formalism interms of a Bogolyubov transformation. If the initial ex-citation is described by a soliton in a zero-point fluctua-tion background the in-phase operator does not have theminimum-uncertainty value, unlike the photon perturba-tion operator, which exhibits Poissonian fluctuations.

The local oscillator pulse shape in the balanced detec-tor orthogonal to the continuum is a superposition of thefunctions sech(x/j) and i@1 2 (x/j)tanh(x/j)#sech(x/j). Ifa hyperbolic secant pulse is used for convenience, cou-pling occurs to the continuum. The squeezing of the con-tinuum is partially coherent with that of the soliton. Thesqueezing of the continuum is oscillatory. In fact, thecontinuum contribution can improve the squeezing be-yond that which is achievable with an LO whose shape isadjusted to be orthogonal to the continuum.

APPENDIX A: CONTINUUM MATRIXELEMENTSThe contribution of the continuum to the normalized dif-ference current is

DQcont

q5 2

1

2 H E dx sech~x !(c,s

E dV

2p

3 Fc,s@exp~2ic 2 it/2!fc,s~x, V, t !

1 exp~ic 1 it/2!fc,s* ~x, V, t !#J5 2

1

2cos c (

c,sE dV

2pFc,s E dx@fI1* ~x, t !

3 fc,s~x, V, t ! 1 fI1~x, t !fc,s* ~x, V, t !#

1i

2sin c (

c,sE dV

2pFc,s E dx@fI1* ~x, t !

3 fc,s~x, V, t ! 2 fI1~x, t !fc,s* ~x, V, t !#. (A1)


The term that multiplies cos c is zero, because it is thestandard projection with the adjoint function fI1(x, t) thatis orthogonal to the continuum. Hence the operatorDQcont can be expressed more simply:

DQcont

q5 E dV

2p@cos cFc~V!Ic~V, t !

1 sin cFs~V!Is~V, t !# (A2)

where

Ic~V, t ! 5 ReH E dx@fI1~x, t !fc* ~x, V, t !#J , (A3a)

Is~V, t ! 5 ImH E dx@fI1~x, t !fs* ~x, V, t !#J . (A3b)

The integrals contain hyperbolic functions with the mul-tiplier exp(6iVx). Therefore the integrals involved inEqs. (A3) can be reduced to simple Fourier transforms ofhyperbolic functions. Using these transforms, one findsthat

Ic~V, t ! 5 0, (A4a)

Is~V, t ! 5 p~1 1 V2!sechS p

2V D cos@~1 1 V2!t/2#. (A4b)

For the evaluation of s22(cont, cont) we need four correla-

tion functions: ûFc,s(V)Fc,s(V8)u&. Consider first thecorrelation function, ûFc(V)Fc(V8)u&:

ûFc~V!Fc~V8!u&

5 1/4K U E dx@D a~x !fIc* ~x, V, 0 ! 1 D a†~x !fIc~x, V, 0 !#

3 E dx8@D a~x8!fIc* ~x8, V8, 0 !

1 D a†~x8!fIc~x8, V8, 0 !#U L5 1/4 E dx E dx8ûD a~x !D a†~x8!u&

3 fIc* ~x, V, 0 !fIc~x8, V8, 0 !

5 1/4F E dxfIc* ~x, V, 0 !fIc~x, V8, 0 !G5

1

4

2p

@1 1 V2#2 d ~V 2 V8!. (A5)

In the same way, one finds for ûFs(V)Fs(V8)u& that

ûFs~V!Fs~V8!u& 5 1/4 E dxfIs* ~x, V, 0 !fIs~x, V8, 0 !

51

4

2p

@1 1 V2#2 d ~V 2 V8!. (A6)

Finally, the cross correlation gives

1/2ûFc~V!Fs~V8! 1 Fs~V8!Fc~V!u&

5 1/8K U E dx@Da~x !fIjc~x, V, 0 !

1 Da†~x !fIc~x, V, 0 !#E dx8@Da~x8!fIs* ~x8, V8, 0 !

1 Da†~x8!fIs~x8, V8, 0 !#U L 1 h.c.

5 1/8 E dx E dx8ûDa~x !Da†~x8!u&

3 fIc* ~x, V, 0 !fIs~x8, V8, 0 ! 1 h.c.

5 1/8F E dxfIc* ~x, V, 0 !fIs~x, V8, 0 !

1 E dxfIs* ~x, V8, 0 !fIc~x, V, 0 !G . (A7)

Now the functions fIc(x, V, 0) and fIs(x, V, 0) differ onlyby the factor i. Hence, cross correlation (A7) is zero. Wefind, for the autocorrelation of the continuum, that

s22~cont, cont! 5 K U E dV

2pFs~V!Is~V, t !

3 E dV8

2pFs~V8!Is~V8, t !U L

51

4

1

2pE dV

Is2~V, t !

~1 1 V2!2

5p

8E dV sech2S p

2V D cos2@~1 1 V2!t/2#.

(A8)Next we turn to the cross-correlation matrix. Again, onlythe 22 element of the matrix is nonzero:

s22~sol, cont! 5 K U2DA2~t ! 5 E dV8

2pFs~V8!Is~V8, t !U L .

(A9)Inasmuch as DA2(t) 5 DA2(0) 1 4F(t)DA1(0), we

must compute the cross correlations between DA1(0) andDA2(0), on the one hand, and Fs on the other hand.Consider first the cross correlation ûDA1(0)Fc(V)u&:

ûDA1~0 !Fs~V!u&

5 1/4K U E dx@Da~x !fI1* ~x !

1 Da†~x !fI1~x !#E dx8@Da~x8!fI* ~x8, V, 0 !

1 Da†~x8!fIs~x8, V, 0 !#U L5 1/4 E dx E dx8ûDa~x !Da†~x8!u&fI1* ~x !fIs~x8, V, 0 !

5 1/4 E dx E dx8d ~x 2 x8!fI1* ~x !fIs~x8, V, 0 !

5 1/4 E dxfI1* ~x !fIs~x, V, 0 !, (A10)


and thus

1/2ûDA1~0 !Fs~V! 1 Fs~V!DA1~0 !u&

5 1/8F E dxfI1* ~x !fIs~x, V, 0 !

1 E dxfI1~x !fIs* ~x, V, 0 !G [ C1s~V!. (A11)

In the same way one finds that

1/2ûDA2~0 !Fs~V! 1 Fs~V!DA2~0 !u&

5 1/8F E dxfI2* ~x !fIs~x, V, 0 !

1 E dxfI2~x !fIs* ~x, V, 0 !G [ C2s~V!. (A12)

Again, one may evaluate the integrals by simple Fouriertransforms. One finds that

C2s 51

4~1 1 V2!2 F2

3p 1

p2

6V tanhS p

2V D

1p2

6V3 tanhS p

2V D GsechS p

2V D , (A13)

where C1s 5 0. Thus we find for the cross-correlationmatrix that

s22~sol, cont! 5 2 E dV

2p@C2s~V! 1 2F~t !C1s~V!#Is~V, t !

5 2 E dV

2pC2s~V!Is~V, t !

51

4E dV

1

~1 1 V2!F2

3p 1

p2

6V tanhS p

2V D

1p2V3

6tanhS p

2V D Gsech2S p

2V D

3 cosF ~1 1 V2!t

2 G . (A14)

ACKNOWLEDGMENTThis research was supported in part by Office of NavalResearch grant N00014-92-J-1302.

H. A. Haus’s e-mail address is [email protected].

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Soliton squeezing and the continuum