January 15, 1999 / Vol. 24, No. 2 / OPTICS LETTERS 89

Soliton squeezing in a highly transmissive nonlinear opticalloop mirror

Dmitry Levandovsky, Michael Vasilyev, and Prem Kumar

Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208-3118

Received August 20, 1998

A perturbation approach is used to study the quantum noise of optical solitons in an asymmetric f iberSagnac interferometer (a highly transmissive nonlinear optical loop mirror). Analytical expressions for thethree second-order quadrature correlators are derived and used to predict the amount of detectable amplitudesqueezing along with the optimum power-splitting ratio of the Sagnac interferometer. We find that it is thenumber-phase correlation owing to the Kerr nonlinearity that is primarily responsible for the observable noisereduction. The group-velocity dispersion affecting the field in the nonsoliton arm of the fiber interferometeris shown to limit the minimum achievable Fano factor. 1999 Optical Society of America

OCIS codes: 270.6570, 060.5530.

Recently, generation of sub-Poissonian light in anasymmetric fiber Sagnac interferometer was pre-dicted1 by use of a numerical technique that relieson a positive-P representation.2 Two groups havevalidated this prediction,3,4 demonstrating 3.9 and5.7 dB of amplitude squeezing, respectively. In thisLetter we clarify the physical mechanism that isresponsible for this noise reduction by deriving ananalytical solution that is based on a perturbationapproach.5,6 We find that the observable amplitudesqueezing in the asymmetric Sagnac loop is mainlydetermined by the strong number-phase correlation,which grows with propagation distance, similar to thecw case studied by Kitagawa and Yamamoto.7 Thissqueezing mechanism differs from that of solitonspectral filtering, wherein the noise reduction re-sults mostly from the correlation between the photonnumber and the bandwidth, which does not growwith distance.6 We calculate the amount of availablesqueezing as well as the optimum power-splitting ratioof the Sagnac interferometer and explain the role ofthe group-velocity dispersion. Our model takes intoaccount the complete contribution of the continuum tothe detected quantum noise.6

We analyze the following experimental configura-tion. A short optical pulse is launched into a fiberSagnac interferometer. The action of the beam split-ter results in two independent modes, an N ­ 1 soli-ton asol propagating in one direction around the fiberloop and a weaker sN2 ,, 1d dispersive pulse agvdpropagating in the other direction. After propaga-tion around the fiber loop, the two pulses interfere atthe same beam splitter, so most of the soliton powerappears in the transmitting arm of the interferome-ter, mixed with a small fraction of the power ofthe dispersive pulse. The transmitted pulse is sub-sequently directed onto a photon-counting detector.For a beam splitter with intensity transmission co-eff icient T the two counterpropagating waves insidethe Sagnac loop are asolsjd ­ fa0 1 Dasolsjdgexpsijy2dand agvdsjd ­ ifs1 2 T dyT g1/2a0exps2icd 1 iDagvdsjd.Here a0svd ­ p sechspvy2d is the fundamental soli-

0146-9592/99/020089-03$15.00/0

ton shape in the frequency domain, j is the normal-ized propagation distance inside the loop in units ofdispersion length, c ; v2jy2 is the quadratic phaseshift that is due to the group-velocity dispersion, andDasol and Dagvd are quantum noises associated withthe soliton and the dispersive pulse, respectively, i.e.,fDaisv, jd, Day

j sv0, jdg ­ 2pdsv 2 v0ddij for i, j [hsol, gvdj. The two noise operators represent inde-pendent coherent states immediately after the beamsplitter sat j ­ 0d: kDaisv, 0dDaj sv0, 0dl ­ 0. In ouranalysis we consider T to be large enough that the non-linear effects in the propagation of the weaker pulseagvd can be neglected, i.e., Dagvdsjd remains in a coher-ent state for all j. The soliton asolsjd, however, evolvestoward a nonclassical state, according to the nonlinearSchrodinger equation.5,6

The output field in the transmitting arm of theinterferometer is given by

aout ; aout 1 Daout ­p

T asolsjd 1 ip

1 2 T agvdsjd ,

(1)

where

aout ­p

T a0 expsijy2dΩ12

1 2 TT

expf2isc 1 jy2dgæ

,

(2)Daout ­

pT Dasolsjdexpsijy2d 2

p1 2 T Dagvdsjd . (3)

Defining the two noise operators Dacsolsv, jd ;

fDasolsv, jd 1 Daysolsv, jdgy2 and Das

solsv, jd ;fDasolsv, jd 2 Day

solsv, jdgy2, which represent am-plitude (cosine) and phase (sine) f luctuations of thesoliton field, and using Eqs. (2) and (3), we find thefollowing Fano factor for ideal direct detection (unitdetection efficiency) at the transmitting output port:

F sjd ­ 1 1 TµZ

jaoutj2dvy2p

∂21 ZZjaoutj ja0

outj

3X

i, j­hc, sjFiG

ijN sv, v0, jdFj 0dvdv0y4p2, (4)

1999 Optical Society of America

90 OPTICS LETTERS / Vol. 24, No. 2 / January 15, 1999

where Fc ­ cos wsvd, Fs ­ 2i sin wsvd, and wsvd ­argfaoutsvdg 2 jy2. Our conventions throughout thisLetter are to use prime symbols to denote functionsof v0, where applicable, and to assume all integralsto be from minus infinity to plus infinity. In Eq. (4)G

ijN sv, v0, jd are the normally ordered correlators,

which are evaluated at length j as

GccN sv, v0, jd ; 4k: Dac

solDac0sol :lj , (5)

GssN sv, v0, jd ; 4k: Das

solDas0sol :lj , (6)

GcsN sv, v0, jd ; 4k: Dac

solDas0sol :lj

­ 2GcspN sv, v0, jd ­ Gsc

N sv0, v, jd . (7)

We can easily obtain these noise correlators byfollowing the perturbation approach described inRef. 6. For the sake of convenience we first evaluatethem in the time domain and then use the Fourier-transform relation to convert them to the form givenin Eqs. (5)–(7). The cosine and sine noise operators inthe time domain can be constructed with the normal-mode expansion as follows:

fDasolst, jd 1 Daysols2t, jdgy2 ­

Xi­n, p, c

DXi , (8)

fDasolst, jd 2 Daysols2t, jdgy2 ­

Xi­u, t, s

DXi , (9)

where the six contributing terms are given byDXi[hn, p, u, tjst, jd ; Visjdfistd and DXi[hc, sjst, jd ;R

VisV, jdfisV, tddVy2p. The time-domain normalmodes used in the expansion of the cosine operator,fnstd ­ f1 2 t tanhstdgsechstd, fpstd ­ 2it sechstd,and fcstd ­ hfsV2 2 1d 2 2iV tanh stdgexps2iVtd 1

2 sech2stdcossVtdjysV2 1 1d, have real Fourier trans-forms, whereas the modes that are pertinent to the sineoperator, fustd ­ 2i sechstd, ftstd ­ tanhstdsechstd,and fsstd ­ ihfsV2 2 1d 2 2iV tanhstdgexps2iVtd 2

2i sech2stdsinsVtdjysV2 1 1d, have imaginary Fouriertransforms. Note that we employ the Heisenbergpicture, where all the j dependence is in the operatorcoefficients Visjd, which are Hermitian and can be as-sociated with either the time- or frequency-dependentmodes. Physically, fn, fu , fp, and ft represent per-turbations to the soliton field owing to changes inphoton number, phase, momentum (frequency), andposition (time), respectively, whereas fc and fs rep-resent amplitude and phase perturbations of thecontinuum, respectively. The equations of motionfor the operator coefficients are found by substitutionof the modal expansion into the linearized nonlinearSchrodinger equation and employment of the orthogo-nality relations described in Refs. 5 and 6. In thisway the inverse Fourier transforms of the correlatorsin Eqs. (5)–(7), in their unordered form, are shownto be

Gccst, t0, jd ­ 4X

i, j­n, p, ckDXistdDXj st0dlj , (10)

Gssst, t0, jd ­ 4X

i, j­u, t, skDXistdDXj st0dlj , (11)

Gcsst, t0, jd ­ 4X

i­n, p, c; j­u, t, skDXistdDXj st0dlj . (12)

We evaluated all the second-order moments betweenthe operator coefficients that are present in Eqs. (10)–(12) for arbitrary values of j. Defining a ; s1 1

V2djy2, b ; sV2 2 V02djy2, and g ; s2 1 V2 1V02djy2, we obtain the following nonzero moments:

kV2n lj ­ 1y2, kV2

u lj ­ sp2 1 12dy72 1 j2y2 , (13)

kV2plj ­ 1y6, kV2

t lj ­ sp2 1 4j2dy24 , (14)"kVnVulj

kVpVtlj

#­

i4

"21

1

#2 j

"1y21y6

#, (15)

266664kVnVcsVdlj

kVnVssVdlj

kVpVcsVdlj

kVpVssVdlj

377775 ­a0sVd

4

2666642cos a

sin a

sVy3dcos a

2sVy3dsin a

377775 , (16)

"kVuVcsVdlj kVtVcsVdlj

kVuVssVdlj kVtVssVdlj

#­

"sin a 2j cos a

cos a j sin a

#

3

"kVuVssVdl0 kVtVssVdl0

kVnVcsVdl0 kVpVcsVdl0

#, (17)

2664 kVcsVdVcsVdlj

kVssVdVssVdlj

kVcsVdVssVdlj

3775 ­

264 1 cos b 2cos g

1 cos b cos g

i sin b sin g

375

3p

2

26666666664

dsV 2 V0d

sV 2 V0d2 1 46sV2 1 1d sV02 1 1d

sV 2 V0dsinhfpsV 2 V0dy2g

sV 1 V0d2 2 6VV0 2 26sV2 1 1d sV02 1 1d

sV 1 V0dsinhfpsV 1 V0dy2g

37777777775,

(18)

kVuVssVdl0 ­ 2a0sVd

4

"2y3

V2 1 11

pV

6tanh

µpV

2

∂#,

(19)

kVtVssVdl0 ­a0sVd

2

"V

V2 1 12

p

4tanh

µpV

2

∂#.

(20)

Note that all the operator coefficients of the same class(i.e., cosine or sine) commute with each other.

We evaluated the integrals that are present inEqs. (10)–(12) numerically, took the Fourier trans-forms of the resulting correlators, and used them inEq. (4) to obtain the Fano factor. The results are plot-ted in Fig. 1 versus the intensity-transmission coef-ficient T and the propagation distance z ; 2jyp insoliton periods. As shown, large amounts of squeezingcan be obtained for propagation distances z . 3. The

January 15, 1999 / Vol. 24, No. 2 / OPTICS LETTERS 91

Fig. 1. Top: contour plot showing the Fano factor, includ-ing dispersion, versus T (solid curves, F ­ 210 dB; dashedcurves, F ­ 25 dB; thick solid curves, F ­ 0 dB; dotted-dashed curves, F ­ 5 dB; dotted curves, F ­ 10 dB; thickgray curves, F ­ 15 dB). Middle: Fano factor minimizedby the proper choice of T with (solid curves) and without(dashed curves) dispersion taken into account; the Fano fac-tor limit imposed by the loss s1 2 T d in each case (dottedcurves) is also shown. Bottom: optimum T correspondingto the minimum Fano factor in the middle figure, with (solidcurves) and without (dashed curves) dispersion. The dis-tance is in soliton periods in all three plots.

noise-reduction regions repeat approximately everyeight soliton periods; propagation over this distance re-sults in the soliton’s acquiring a nonlinear phase shiftDjy2 ­ 2p with respect to the weak pulse. The firstthree local minima of the Fano factor are found to be210.6 dB sz ø 5.5, T ø 0.92d, 212.7 dB sz ø 13.2, T ø0.95d, and 213.7 dB sz ø 21.1, T ø 0.96d.

The periodicity in the Fano factor indicates that thephysical mechanism behind the noise reduction in aSagnac interferometer is very similar to those of thecw field in a nonlinear Mach-Zehnder interferometer7

and of soliton quadrature squeezing.5 As the solitonpropagates in one direction around the loop, one ofthe field quadratures becomes squeezed owing to self-phase modulation. This squeezing is manifested bythe correlations of Vn, Vc, with Vu, Vs. Although theamount of squeezing increases with j, the angle of thesqueezed quadrature with respect to the soliton meanfield approaches zero. The amplitude quardrature,however, remains unsqueezed. By mixing a small co-herent mean-field component with the squeezed soli-ton field, one can rotate the output mean field aout toselect the minimum noise quadrature7,8 while introduc-ing only a small amount of loss to the squeezed compo-nent. The Sagnac interferometer performs the mixingof the two fields with the relative phase shift deter-mined by the nonlinear phase of the soliton inside thefiber loop. This configuration provides good stabilityof the interference while limiting the minimum Fanofactor by only the amount of the beam-splitter loss,1 2 T . To illustrate this point we calculated the op-timum noise reduction for the case in which the disper-sion in the weaker pulse is turned off [c in Eq. (2) is set

to 0]. In Fig. 1 we compare the Fano factors with andwithout dispersion in the weaker pulse with the limitsset by the corresponding beam-splitting losses, 1 2 T .One can see that the amount of available quadraturesqueezing inside the loop is large enough to make 1 2 Tthe main constraint on the observable noise reduc-tion. For long propagation distances the fundamentalsoliton a0std ­ sechstd is close to the matched local-oscillator shape for detection of quadrature squeez-ing.5 Therefore, since the undispersed pulse is bettermatched to the squeezed mode than is the dispersedpulse, the amount of mean field needed to rotate theoutput field is smaller, resulting in lower necessaryloss, 1 2 T . As z increases, the optimum transmissionapproaches unity, and the Fano factor becomes arbi-trarily small in an increasingly narrow range of valuesof T . On the other hand, when the dispersion of theweaker pulse is taken into account, the optimum trans-mission becomes smaller and the requirement on itstolerance relaxes, whereas the achievable noise reduc-tion suffers. For longer propagation distances (up toz ­ 50), we find that the Fano factor slowly approachesa value of F ø 216 dB. Note that in the time domaindispersion also leads to a mean phase shift of agvd withrespect to asol, causing regions of optimum squeezingto slide toward shorter z by approximately one solitonperiod.

Although our model does not provide numericalestimates when the pulse propagating in the strongarm of the interferometer is not a soliton sN fi 1d,one can make approximate predictions by shifting theranges of squeezing in Fig. 1 along the z direction bythe amount of the additional nonlinear phase shift.Such predictions are found to be in agreement withnumerical1,3 and experimental3,4 results.

The authors acknowledge useful discussions with A.Mecozzi. This work was supported in part by the U.S.Office of Naval Research and the National ScienceFoundation.

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