January 1, 1999 / Vol. 24, No. 1 / OPTICS LETTERS 43 Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering Dmitry Levandovsky, Michael Vasilyev, and Prem Kumar Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208-3118 Received June 11, 1998 We study the quantum-noise properties of spectrally filtered solitons in optical fibers. Perturbation theory, including a quantum description of the continuum, is used to derive a complete analytical expression for the second-order correlator of the amplitude quadrature. This correlator is subsequently used to optimize the frequency response of the filter numerically in order to achieve the minimum photon-number noise. For propagation distances up to three soliton periods, the length at which the best noise reduction occurs, a square filter is found to be approximately optimum. For longer distances, more-complicated filter shapes are predicted for the best noise reduction. 1999 Optical Society of America OCIS codes: 270.6570, 060.5530. Recently, sub-Poissonian light was generated by frequency filtering of solitons after their propaga- tion through the f iber. 1 Optical solitons were launched into the fiber, and the light emerging from the spectral filter was directed onto a photon-counting detector, where its noise was measured. Numerical and analytical models of these experiments were developed, based on a positive P representation 2 and a backpropagation approach. 3 One critical issue that was not addressed in these models is that of an optimum f ilter that maximizes the observable quantum-noise reduction for a given length of fiber. In this Letter we offer a solution to the matched-filter problem. We derive, for the first time to our knowl- edge, an analytical expression for the quantum-noise correlator of the soliton amplitude quadrature as a function of the fiber length, taking into account the complete contribution of the continuum. 4 An experi- mental study of this correlator was recently reported. 5 Propagation of the mode-amplitude operator ˆ ast, jd inside an optical f iber is described by the quantum non- linear Schr¨ odinger equation (NLSE), which in dimen- sionless coordinates is ≠ ≠j ˆ ast, jd › i ∑ 1 2 ≠ 2 ≠t 2 1 ˆ a y st, jd ˆ ast, jd ∏ ˆ ast, jd . (1) If the operators in this equation are replaced by c-numbers, Eq. (1) is known to have the classical fundamental-soliton solution ast, jd › f n stdexpsijy2d; sechstdexpsijy2d, given here in the canonical form corresponding to the average number of photons k ˆ N l › 2, where ˆ N ; R ˆ a y std ˆ astd dt. To circumvent the problem of solving for the nonlinear evolution of operator ˆ a, we use the soliton perturbation approach developed in Ref. 6. We write the operator in the following form: ˆ ast, jd › f n stdexpsijy2d 1D ˆ bst, jd ;f f n std 1D ˆ ast, jdgexpsijy2d , (2) with D ˆ a subject to the usual commutation relations everywhere inside the fiber: fD ˆ ast, jd, D ˆ ast 0 , jdg › fD ˆ a y st, jd, D ˆ a y st 0 , jdg › 0, fD ˆ ast, jd, D ˆ a y st 0 , jdg › dst 2t 0 d. Substituting Eq. (2) into Eq. (1) and keeping only terms that are linear in D ˆ b divides the analysis of Eq. (1) into two separate problems: the soliton solution of the classical NLSE and the solution of the linearized operator equation ≠D ˆ b ≠j › i 2 ≠ 2 ≠t 2 D ˆ b 1 2ijast, jdj 2 D ˆ b 1 iast, jd 2 D ˆ b y , (3) where the quantum-mechanical fluctuation operator D ˆ b represents perturbation of the classical soliton f ield by the quantum noise. This linearization approach is valid if the photon-number noise is small compared with the mean number of photons in the soliton, which is usually the case in most experiments. In the linearization approximation, because we re- tain terms only up to first order in D ˆ b, the photon- number noise is determined by the fluctuations in the amplitude quadrature. We assume that the filter H svd at the output of the fiber is linear. Because the filtered light is direct-detected, we can disregard the phase factor expsijy2d in Eq. (2), for both the noise and the mean fields, as well as the phase of H svd. How- ever, we must restrict the filter transfer function such that 0 # jH svdj # 1 for it to represent a physically real- izable filter. In the frequency domain, the f luctuation operator after the f ilter is D ˆ a o sv, jd › jH svdjD ˆ asv, jd 1 q 1 2 jH svdj 2 ˆ vsvd , (4) where ˆ v is a vacuum-state operator that describes the frequency-dependent loss of the filter. By normalizing the photon-number variance to the average output photon number we obtain the Fano factor: F sjd;kD ˆ N o sjd 2 lyk ˆ N o l › 1 1 k ˆ N o l 21 ZZ dv 2p dv 0 2p f n svd jH svdj 2 G N sv, v 0 , jd jH sv 0 dj 2 f n sv 0 d , (5) where k ˆ N o l; R jH svdj 2 f n svd 2 dvy2p is the aver- age number of photons at the f ilter output and G N sv, v 0 , jd is the normally ordered part of the second-order amplitude-quadrature correlator: 0146-9592/99/010043-03$15.00/0 1999 Optical Society of America
Perturbation theory of quantum solitons: continuum evolutionand optimum squeezing by spectral filtering
Dmitry Levandovsky, Michael Vasilyev, and Prem Kumar
Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208-3118
Received June 11, 1998
We study the quantum-noise properties of spectrally filtered solitons in optical f ibers. Perturbation theory,including a quantum description of the continuum, is used to derive a complete analytical expression forthe second-order correlator of the amplitude quadrature. This correlator is subsequently used to optimizethe frequency response of the filter numerically in order to achieve the minimum photon-number noise. Forpropagation distances up to three soliton periods, the length at which the best noise reduction occurs, asquare filter is found to be approximately optimum. For longer distances, more-complicated filter shapes arepredicted for the best noise reduction. 1999 Optical Society of America
OCIS codes: 270.6570, 060.5530.
Recently, sub-Poissonian light was generated byfrequency filtering of solitons after their propaga-tion through the fiber.1 Optical solitons werelaunched into the fiber, and the light emerging fromthe spectral filter was directed onto a photon-countingdetector, where its noise was measured. Numericaland analytical models of these experiments weredeveloped, based on a positive P representation2 anda backpropagation approach.3 One critical issuethat was not addressed in these models is that ofan optimum filter that maximizes the observablequantum-noise reduction for a given length of fiber.In this Letter we offer a solution to the matched-filterproblem. We derive, for the first time to our knowl-edge, an analytical expression for the quantum-noisecorrelator of the soliton amplitude quadrature as afunction of the fiber length, taking into account thecomplete contribution of the continuum.4 An experi-mental study of this correlator was recently reported.5
Propagation of the mode-amplitude operator ast, jdinside an optical fiber is described by the quantum non-linear Schrodinger equation (NLSE), which in dimen-sionless coordinates is
≠
≠jast, jd i
∑12
≠2
≠t2 1 ayst, jdast, jd∏ast, jd . (1)
If the operators in this equation are replacedby c-numbers, Eq. (1) is known to have theclassical fundamental-soliton solution ast, jd f
nstdexpsijy2d ; sechstdexpsijy2d, given here in the
canonical form corresponding to the average numberof photons kN l 2, where N ;
Raystdastd dt. To
circumvent the problem of solving for the nonlinearevolution of operator a, we use the soliton perturbationapproach developed in Ref. 6. We write the operatorin the following form:
ast, jd fnstdexpsijy2d 1 Dbst, jd
; ffnstd 1 Dast, jdgexpsijy2d , (2)
with Da subject to the usual commutation relationseverywhere inside the fiber: fDast, jd, Dast0, jdg fDayst, jd, Dayst0, jdg 0, fDast, jd, Dayst0, jdg
0146-9592/99/010043-03$15.00/0
dst 2 t0d. Substituting Eq. (2) into Eq. (1) andkeeping only terms that are linear in Db divides theanalysis of Eq. (1) into two separate problems: thesoliton solution of the classical NLSE and the solutionof the linearized operator equation
≠Db≠j
i2
≠2
≠t2 Db 1 2ijast, jdj2Db 1 iast, jd2Dby, (3)
where the quantum-mechanical f luctuation operatorDb represents perturbation of the classical soliton fieldby the quantum noise. This linearization approach isvalid if the photon-number noise is small comparedwith the mean number of photons in the soliton, whichis usually the case in most experiments.
In the linearization approximation, because we re-tain terms only up to first order in Db, the photon-number noise is determined by the f luctuations inthe amplitude quadrature. We assume that the filterH svd at the output of the fiber is linear. Because thefiltered light is direct-detected, we can disregard thephase factor expsijy2d in Eq. (2), for both the noise andthe mean fields, as well as the phase of H svd. How-ever, we must restrict the filter transfer function suchthat 0 # jH svdj # 1 for it to represent a physically real-izable filter. In the frequency domain, the f luctuationoperator after the filter is
Daosv, jd jH svdjDasv, jd 1
q1 2 jH svdj2 vsvd , (4)
where v is a vacuum-state operator that describes thefrequency-dependent loss of the filter. By normalizingthe photon-number variance to the average outputphoton number we obtain the Fano factor:
F sjd ; kDNosjd2lykNol 1 1 kNol21ZZ dv
2p
dv0
2p
fnsvd jH svdj2GN sv, v0, jd jH sv0dj2f
nsv0d , (5)
where kNol ;R
jH svdj2fnsvd2dvy2p is the aver-
age number of photons at the filter output andGN sv, v0, jd is the normally ordered part of thesecond-order amplitude-quadrature correlator:
Note that all integrals in Eq. (5) and throughout thisLetter are assumed to have limits from 2` to `.Computationally, it is easier to work with a two-dimensional Fourier transform of Gsv, v0, jd:
The solution of Eq. (3) can be written as a normalmode expansion6:
Dast, jd Z dV
2pfVcsV, jdfcsV, td 1 VssV, jdfssV, tdg
1X
in, p, t, u
Visjdfistd , (6)
wherein the four discrete modes fn, fp, ft, and fu
represent perturbations to the soliton shape thatare due to changes in photon number, momentum(frequency), position (time), and phase, respectively,and fc and fs are the symmetric and antisymmet-ric parts, respectively, of the modes that representperturbations of the continuum (dispersive radia-tion) in the fiber. In our convention, the symmetricmodes satisfy f std fps2td and are given by fn f1 2 t tanhstdgsechstd, fp 2it sechstd, and fc hfsV2 2 1d 2 2iV tanhstdgexps2iVtd 1 2 sech 2std 3
cossVtdjysV2 1 1d, which are real in the frequencydomain. The antisymmetric [i.e., f std 2fps2td]modes, ft tanhstdsechstd, fu 2i sechstd, and fs ihfsV2 2 1d 2 2iV tanhstdgexps2iVtd 2 2i sech2std 3sinsVtdjysV2 1 1d, are imaginary in v.
Note that the j dependence in Eq. (6) is associatedwith the operator coefficients Vi, Vc, and Vs (Hei-senberg picture), which are Hermitian. The whitecoherent-state quantum noise at the input of the fibersj 0d perturbs all the modes of the linearized NLSE,as shown by the expansion in Eq. (6). The operatorcoefficients in Eq. (6) then propagate through to theend of the fiber, where the noise is reconstructed, onceagain, by superposition of all the normal modes.
By substituting the expansion in Eq. (6) into Eq. (3)one can show that the photon number and momentumdo not change as the soliton propagates along the fiber,i.e., Vnsjd Vns0d and Vpsjd Vps0d, whereas the timeand phase evolve according to Vtsjd Vts0d 2 jVps0dand Vusjd Vus0d 2 jVns0d. Similarly, for the twocontinuum operators VcsV, jd VcsV, 0dcosfs1 1
V2djy2g 1 VssV, 0dsinfs1 1 V2djy2g and VssV, jd VssV, 0dcosfs1 1 V2djy2g 2 VcsV, 0dsinfs1 1 V2djy2g.Operators Vc and Vs are essentially the quadra-ture operators associated with a continuum modeV and subject to the usual commutation relations:fVcsV, jd, VssV0, jdg ipdsV 2 V0d and fVcsV, jd,VcsV0, jdg fVssV, jd, VssV0, jdg 0. Note that ourcontinuum mode functions are consistent with thedefinition in Ref. 6, although we have adopted a formthat is convenient for Heisenberg representation.
To project out the expansion coefficients in Eq. (6)one needs to construct an orthogonality relation bypairing the growing solutions with the decaying(adjoint) ones such that the cross energy is con-served.6 The adjoint solutions, shown here withunderbars, are solutions of the adjoint linearizedNLSE, which differs from Eq. (3) by the sign of theDby term, and are related to the solutions of Eq. (3).Specif ically: f
u 2ifn, f
t ifp, f
s ifc, f
n ifu ,
fp
2ift, fc
2ifs [note that fn
is the same as thatin Eq. (2)]. Defining the scalar product as
k fi ? fjpl Re
Zfistdf
jpstddt
ReZ
fisvdfjpsvd
dv
2p, (7)
we obtain the orthogonality conditions: k fi ?
fjpli, j[hn, p, t, u, c, sj Dij , where Dij dij in all
cases except when i j [ hc, sj, in which caseDii 2pdsV 2 V0d.
Only the symmetric (real in the frequency domain)modes contribute to the amplitude-quadrature cor-relator. Accordingly, we define two time-domainquadrature-like operators, Dacst, jd fDast, jd 1
Days2t, jdgy2 and Dasst, jd fDast, jd 2 Days2t,jdgy2, where only the symmetric quadrature Dacst, jdis needed for our calculation. The quantum-f luctuation operators are projected out by use ofEq. (7), i.e., Vc, n, p ; kDac ? f
c, n, ppl and Vs, u, t ;
kDas ? fs, u, t
pl. This projection allows us to obtain ananalytical expression for the second-order quadraturecorrelator in the time domain:
Fig. 1. Top, gray-scale visualization of the optimum filterresponse jH svd j2opt, where white and black correspond to0 and 1, respectively. Bottom, Fano factor obtained for asquare filter with a 0.18 (dashed curve) and the optimumfilter (solid curve); total loss a for the square filter (dotted–dashed line) and the optimum filter (dotted curve). Inboth plots distance is in soliton periods.
Fig. 2. Top, normally ordered quadrature-noise correlatorGN sv, v0, jd. Bottom, frequency response jH svd j2opt of theoptimum filter (interpolated circles), amplitude-quadraturevariance GN sv, v, jd (dotted curve), and zero-frequency-to-sideband correlation function GN s0, v, jd (dashed curve).All functions are shown for z ; 2jyp 10.73.
and kVn2l 1y2, kVp
2l 1y6.We have numerically evaluated the integrals in the
correlator in Eq. (8) and taken its Fourier transformto get to a form that is useful for the optimumfrequency-filtering problem. Note that, in the absenceof filtering, only the kVn
2l term contributes to theFano factor, making it equal to 1 because, owing toorthogonality, all the other terms integrate to zero inEq. (5). This fact makes the role of the filter clear, asit permits mixing in of the negatively correlated termsin Eq. (8). The noise reduction takes place mainlybecause of the kVnVcl term, which describes the factthat an increasing photon number causes an increasedsoliton bandwidth, thereby resulting in a higher loss
introduced by the spectral filter. We observe herethat the physical symmetry of the problem suggests anoptimum filter function that is even in the frequencydomain. Hence the terms that contain fp do notcontribute to the optimally filtered noise.
With the form of the noise correlator in hand,we are able to find the optimum filter frequencyresponse jH svdj2opt by minimizing the Fano factor.Because of the constraint that 0 # jH svdj2 # 1, analyti-cal optimization by variational methods is impractical.Instead, we utilized a numerical quasi-Newton con-strained optimization algorithm. The resultant filtershape along with the corresponding Fano factor andthe filter loss a ; f1 2 kNolykN lg is shown in Fig. 1as a function of the propagation distance z ; 2jypin soliton periods. We have also plotted the resultsfor a square filter with its transmission bandwidthadjusted sa ø 0.18d for maximum noise reduction atz 3. For this filter our results are in an excel-lent agreement with those in Ref. 3. As can be seenfrom Fig. 1, the best noise reduction of ø6.5 dB isachieved for z ø 3, where the optimum filter is closeto a square shape. The effect of optimization becomesevident for z . 3, where the optimum filter acquiresmore-complicated shapes to take advantage of the fastoscillations that develop in the continuum part of thecorrelator GN sv, v 0, jd, as illustrated in Fig. 2.
In conclusion, it is mainly the correlations betweenthe continuum and the soliton photon number that leadto the quantum-noise reduction after spectral filtering.We found the optimum filter that establishes a theoret-ical limit on the observable noise reduction. For thefiber lengths up to three soliton periods, at which thebest noise reduction occurs, the square filter is shownto be a good approximation to the optimum filter. Forlonger lengths, however, the optimum filter shape de-velops a number of sidebands, resulting in a clear im-provement over the square filter.
The authors acknowledge useful discussions with A.Mecozzi and J. Nocedal. This work was supported inpart by the U.S. Office of Naval Research.
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