Polarization-squeezed light formation in a medium with electronic Kerr nonlinearity

0.17 MB·English

by  F. Popescu

Polarization-squeezed light formation in a medium
with electronic Kerr nonlinearity
∗
F. Popescu
Physics Department, Florida State University, Tallahassee, Florida, 32306
(January 12, 2004)
4
0 Abstract
0
2

n
We analyze the formation of polarization-squeezed light in a medium with
a
J electronic Kerr nonlinearity. Quantum Stokes parameters are considered and

2
the spectra of their quantum fluctuations are investigated. It is established
1
that the frequency at which the suppression of quantum fluctuations is the

2 greatest can be controlled by adjusting the linear phase difference between
v
2 pulses. Weshownthatbyvaryingtheintensityorthenonlinearphaseshiftper
7 photon for one pulse, one can effectively control the suppression of quantum
0
fluctuations of the quantum Stokes parameters.
4
0 PACS: 42.50.Dv, 42.50.Lc.
2
0
Keywords: ultrashort light pulse, self- and cross-phase modulation, electronic Kerr non-
/
h
linearity, quantum Stokes parameters.
p
-
t
n
a I. INTRODUCTION
u
q
: Knowledgeofthequantumpropertiesofthepolarizationoflightisessentialwhentreating
v
i issues concerning Einstein-Podolski-Rosen (EPR) paradox [1] or Bell’s inequality [2]. The
X
quantum analysis ofthelightpolarizationisgenerallybased onStokes parameters associated
r
a withHermitianStokesoperators[3,4]. Wecallaquantumstatetobeapolarization-squeezed
(PS)stateifithasthelevel ofquantumfluctuationsofoneofthequantumStokesparameters
smaller than the level corresponding to the coherent state [5]. Such a state can be generated
by using the optical parametric amplification (χ2) [6,7]. Recently, effective methods of PS
state formation using Kerr-like media (χ3) and optical solitons in fibers have been proposed
[5,8–11]. IntheKerrmediumtheself-phasemodulation(SPM)occurs, leadingtoquadrature
squeezing with preservation of the photon statistics [12].
It was noted in [13] that a quantum treatment of SPM of ultrashort light pulses (USPs)
must account for the additional noise related to the nonlinear absorbtion. The study of the
nonlinear propagation in Raman active media [14] accounts for the quantum and thermal
noises asa fluctuating additiontothe relaxationnonlinearity intheinteraction Hamiltonian.
∗
E-mail: florentin [email protected]
1
However, if we deal with USPs propagation, for instance through fused-silica fibers, the
electronic motion on ∼ 1 fs time scale contributes with about 80% to the Kerr effect while
Raman oscillators give only 20% contribution [15]. An attempt to develop the quantum
theory of pulse SPM has been undertaken in [16], where the electronic Kerr nonlinearity is
modelled as a Raman-like one. The electronic nonlinearity was considered correctly with
the interaction Hamiltonian [17] and the momentum operator [18] in the normally ordered
form.
If two-mode radiation with orthogonal polarization and/or different frequencies prop-
agates through a Kerr medium, then the cross-phase modulation (XPM) and parametric
interaction can also occur. The approach in [17,18] has been extended for the combined
SPM–XPM of USPs in [19], by considering the spectra of fluctuations of quadratures. No-
′′
tice that, for interacting solitons in a Kerr medium with anomalous dispersion (k < 0), the
XPMinducesthetransientphoton-numbercorrelationswhileSPManswersfortheintrapulse
ones [20].
The experimental realizationofthepolarizationsqueezing implies thespatialoverlapping
of an orthogonally polarized strong coherent beam with squeezed vacuum on a 50 : 50
beamsplitter [21]ortheinterference oftwoindependent quadrature-squeezed USPsproduced
in a fiber Sagnac interferometer [8]. The last series of experiments [10,11] involve both the
temporal and spatial overlaps of two orthogonally polarized quadrature-squeezed USPs.
From the theoretical point of view, the quantum fluctuations of Stokes parameters in the
case of the nonlinear propagation in an isotropic Kerr medium were introduced by the
study of the light polarization [4]. The time-independent quantum treatment of two-mode
interaction [5] predicted the possibility of forming the PS state in anisotropic Kerr media.
In this letter we report on the theoretical computation of the spectra of fluctuations of
the Stokes parameters in the case of two orthogonally polarized propagating USPs in an
anisotropic electronic Kerr medium. We employ simultaneously the SPM for quadrature
squeezing and the XPM for the PS state formation and its control. The estimations are
done in the frame of the quantum theory for SPM-XPM of USPs developed in [19]. The
response time of the electronic Kerr nonlinearity is accounted and the dispersion of linear
properties is described in the first approximation of the dispersion theory. We begin with a
brief description of the model introduced in [19] and based on the momentum operators for
the pulse fields. For consistency, we present some elements of the algebra of time-dependent
Bose operators. By using such algebra we get the average values, the correlation functions,
andthe spectra of quantum fluctuations ofStokes parameters. Our concluding remarks close
the paper.
II. QUANTUM THEORY OF COMBINED SPM-XPM EFFECT IN
ELECTRONIC KERR MEDIUM
The quantum theory developed in [19] is based on the following momentum operators:
∞ t
Gˆ(j) (z) = h¯β dt H(t−t )Nˆ[nˆ (t,z)nˆ (t ,z)]dt , (1)
spm jZ−∞ Z−∞ 1 j j 1 1
∞ t
Gˆ (z) = h¯β˜ dt H(t−t )[nˆ (t,z)nˆ (t ,z)+nˆ (t ,z)nˆ (t,z)]dt , (2)
xpm Z−∞ Z−∞ 1 1 2 1 1 1 2 1
2
˜
where j is the pulse number, β and β are nonlinear coefficients (β = h¯ω k n /8n V
j j 0 0 2(j) 0
[19,22])(j = 1,2),Nˆ isthenormalordering operator,H(t)isthenonlinearresponse function
[H(t) 6= 0 at t ≥ 0 and H(t) = 0 at t < 0], nˆ (t,z) = Aˆ+(t,z)Aˆ (t,z) is the “photon
j j j
number density” operator in the cross-section z of the medium, and Aˆ (t,z) and Aˆ+(t,z)
j j
are the photon anihilation and creation Bose operators with the commutation relations
[Aˆ (t ,z),Aˆ+(t ,z)] = δ δ(t −t ). The thermal noise is neglected and the expressions
j 1 k 2 jk 2 1
(1) and (2) are averaged over the thermal fluctuations. We assume that the Kerr effect
is produced by the electronic motion with characteristic time τ ≤ 1 fs. In our approach
r
the pulse duration τ is much greater than the relaxation time τ , and the Kerr medium is
p r
lossless and dispersionless; that is, the pulses frequencies are off resonance.
In general, the induced third-order polarization P(3) of the medium at the frequency ω
(3) (3) ∗ (3) ∗ (3) ∗
is: P (ω) = [χ E (ω)E (ω)E (ω)+χ E (ω)E (ω)E (ω)+χ E (ω)E (ω)E (ω)]
i j 1122 i j j 1212 j i j 1221 j j i
[5,23]. Since Pthe first two terms in P(3) involve operator products accounted in (1) and
(2), the last term introduces products of the form Aˆ+2Aˆ2 + Aˆ+2Aˆ2 which are neglected
1 2 2 1
since they are connected with the parametrical interaction. The conditions (also assumed
here) under which such terms can be neglected were discussed in [5], where it was noted
that the parametric interaction can be neglected if ∆z ≫ 1 and the phase mismatch ∆ =
2(k −k ) is large, namely |∆| > min{β˜|A (0)|,β˜|A (0)|}. Here β˜ = 8πω2(χ(3) +χ(3) +
2 1 1 2 1122 1212
χ(3) )/c2(k +k ), and k = (ω/c)n . Note that for circularly polarized modes the last term
1221 1 2 j j
in P(3) is absent, which is valid in isotropic Kerr media where modes obey only SPM and
XPM. For elliptically polarized fields the last term in P(3) induces the circular birefringence
∆nc = −(2π/n0)2χ(132)21(|E+(ω)|2−|E−(ω)|2) [23]. However, for the successful experimental
overlapping of the two quadrature-squeezed USPs a birefringence compensator is required
[10,11]. In general, ∆n can be neglected when USPs propagate along some symmetry
c
directions [23].
Since in our case the Kerr effect is of an electronic origin, in the absence of one- and
two-photon and Raman resonances, H(t) can be chosen as H(t) = (1/τ )exp(−t/τ ) at
r r
t ≥ 0 [22]. Then with (1) and (2) the space-evolution equation for Aˆ (t,z) in the moving
1
′ ′ ′
coordinate system (z = z , t = t − z/u, where t is the running time and u is the group
velocity),
∂Aˆ1(t,z) −[Oˆ (t)+Oˆ˜ (t)]Aˆ (t,z) = 0, (3)
1 2 1
∂z
has the solution given by
Aˆ (t,z) = eOˆ1(t)+Oˆ˜2(t)Aˆ (t). (4)
1 0,1
Here Oˆ (t) = iγ q[nˆ (t)], Oˆ˜ (t) = iγ˜q[nˆ (t)], γ = β z, γ˜ = β˜z, nˆ (t) = Aˆ+ (t)Aˆ (t)
j j 0,j j 0,j j j 0,j 0,j 0,j
is the “photon number density” operator at the entrance into the medium z = 0, and
q[nˆ (t)] = ∞ h(t )nˆ (t−t )dt , where h(t) = H(|t|). Aˆ (t,z) can be derived by changing
0,j −∞ 1 0,j 1 1 2
the indexesR1 ↔ 2 in (4). In comparison with the so-called nonlinear Schr¨odinger equation,
used in the quantum theory of optical solitons (see [20] and refs. therein), in (3) the pulse
dispersion spreading in the medium is absent. This approach corresponds to the first ap-
proximation of the dispersion theory [22]. Note that the structure of q[nˆ (t)] is like that
0,j
3
of the linear response in the quantum description of a USP spreading in the second-order
approximation.
The statistical features of pulses at the output of the medium can be evaluated by using
the algebra of time-dependent Bose operators [17–19]. In such algebra we have
Aˆ (t )eOˆj(t2) = eOˆj(t2)+Dj(t2−t1)Aˆ (t ), Aˆ (t )eOˆ˜j(t2) = eOˆ˜j(t2)+D˜(t2−t1)Aˆ (t ), (5)
0,j 1 0,j 1 0,j 1 0,j 1
where D (t − t ) = iγ h(t − t ), D˜(t − t ) = iγ˜h(t − t ), 2γ is the nonlinear phase
j 2 1 j 2 1 2 1 2 1 j
shift per photon for the j-th pulse, and γ˜ is the nonlinear coupling coefficient. By using the
theorem of normal ordering [17,18], one obtains the average values of the Bose operators
over the initial coherent summary state |α (t)i = |α (t)i⊗|α (t)i:
0 0,1 0,2
heOˆj(t)i = eiφj(t)−µj(t), heOˆ˜j(t)i = eiφ˜j(t)−µ˜j(t), (6)
heOˆj(t1)+Oˆj(t2)i = ei[φj(t1)+φj(t2)]−[µj(t1)+µj(t2)]−Kj(t1,t2), (7)
heOˆ˜j(t1)+Oˆ˜j(t2)i = ei[φ˜j(t1)+φ˜j(t2)]−[µ˜j(t1)+µ˜j(t2)]−K˜j(t1,t2). (8)
The parameters φ (t) = 2γ n¯ (t), µ (t) = γ2n¯ (t)/2 are connected with SPM of the
j j 0,j j j 0,j
j-th pulse and φ˜ (t) = 2γ˜n¯ (t), µ˜ (t) = γ˜2n¯ (t)/2 with the pulses’ XPM. Here φ (t)
j 0,j j 0,j j
˜
(φ (t)) is the nonlinear phase addition caused by SPM (XPM). The eigenvalue α (t) of
j 0,j
ˆ
the annihilation operator A (t) over the coherent state |α (t)i can be written as α (t) =
0,j 0,j 0,j
|α (t)|eiϕj(t), where ϕ (t) is the linear phase of the j-th pulse. Then |α (t)|2 = hnˆ (t)i ≡
0,j j 0,j 0,j
n¯ (t). For simplicity let n¯ (t = 0) ≡ n¯ . We separate the time dependence in |α (t)| by
0,j 0,j 0,j 0,j
introducing the pulse’s envelope r (t) so that |α (t)| = |α (0)|r (t) with r (0) = 1. In (7)-
j 0,j 0,j j j
(8), K (t ,t ) = µ r2(t +τ/2)g(τ) and K˜ (t ,t ) = µ˜ r2(t +τ/2)g(τ) are the temporal
j 1 2 0,j j 1 j 1 2 0,j j 1
correlators, where µ = γ n¯ , µ˜ = γ˜n¯ , g(τ) = (1+|τ|/τ )h(τ), and τ = t −t .
0,j j 0,j 0,j 0,j r 2 1
III. QUANTUM STOKES PARAMETERS AND THEIR AVERAGE VALUES,
POLARIZATION DEGREE
In classical optics the polarization state is visualized as a Stokes vector on the Poincar´e
sphere and is characterized by the four classical Stokes parameters {S } . Since S
k k=0,3 0
defines the total intensity of pulse field, {S } characterize light polarization and form a
k k=1,3
Cartesian axis system. Each point on the sphere corresponds to a definite polarization state,
whosevariationischaracterized bythemotionofthepointonthesphere. Inquantumoptics,
ˆ ˆ ˆ ˆ ˆ ˆ
{S } arereplacedbytheoperators{S } whichobey: [S ,S ] = 0,[S ,S ] = 2iε S .
k k=0,3 k k=0,3 0 i i j ijk k
We define the quantum Stokes parameters as: Sˆ (t,z) = 2 nˆ (t,z), Sˆ (t,z) = nˆ (t,z)−
0 j=1 j 1 1
nˆ (t,z), P
2
Sˆ (t,z) = Aˆ+(t,z)Aˆ (t,z)+Aˆ+(t,z)Aˆ (t,z), (9)
2 2 1 1 2
Sˆ (t,z) = i [Aˆ+(t,z)Aˆ (t,z)−Aˆ+(t,z)Aˆ (t,z)], (10)
3 2 1 1 2
where in our case Aˆ (t,z) is given by (4) and Aˆ+(t,z) is its Hermitian conjugate. By using
j j
Eqs. (5)-(6) we compute the average values of {Sˆ (t,z)} on the coherent summary state
k k=0,3
|α (t)i. Finally, we get: hSˆ (t,z)i = 2 n¯ (t), hSˆ (t,z)i = n¯ (t)−n¯ (t), and
0 0 j=1 0,j 1 0,1 0,2
P
4
hSˆ (t,z)i = 2[n¯ (t)n¯ (t)]1/2e−[∆1(t)+∆2(t)]cos[Φ˜ (t)−Φ˜ (t)], (11)
2 0,1 0,2 2 1
hSˆ (t,z)i = 2[n¯ (t)n¯ (t)]1/2e−[∆1(t)+∆2(t)]sin[Φ˜ (t)−Φ˜ (t)], (12)
3 0,1 0,2 2 1
where∆ (t) = µ (t)+µ˜ (t), Φ˜ (t) = φ (t)−φ˜ (t)+ϕ (t). Bymeasurements, oneobtainsaset
j j j j j j j
ofmeasurablequantitiesassociatedwiththeoperators(9)and(10). However, thepresenceof
quantum fluctuations results in an uncertainty for the measured quantities. The fluctuation
uncertainty of each hSˆ (t,z)i can be associated with a particular region of uncertainty on
k
the Poincar´e sphere. In the case of the nonlinear propagation, the ball-region of uncertainty,
specifically for the coherent state of light, changes to the ellipsoid of uncertainty [5].
In classical optics the polarization degree P is the ratio of the intensity of the polarized
part of the radiation I to the total intensity I , and is connected with the classical
pol tot
Stokes parameters by P = R/hS i, where R is the radius of the classical Poincar´e sphere,
0
R2 = 3 hS i2. Since for completely polarized light P = 1, for partially polarized light
k=1 k
0 < PP< 1. The definition of the quantum polarization degree follows the classical one:
P(t,z) = R(t,z)/hSˆ (t,z)i, where R2(t,z) = 3 hSˆ (t,z)i2. With the expressions (11)-
0 k=1 k
(12) we have P
P(t,z) = {1−4n¯ (t)n¯ (t)[n¯ (t)+n¯ (t)]−2(1−e−2[∆1(t)+∆2(t)])}1/2. (13)
0,1 0,2 0,1 0,2
Since in real-life situations ∆ (t)+∆ (t) ≪ 1, we have P(t,z) ≈ 1. Indeed, a recent study
1 2
[3,5] investigated the nonlinear behavior of P(t,z) for the nonclassical states of light and
revealed that the deviation of P(t,z) from 1 is a pure quantum effect. Note that the results
(11)-(13) are similar to the ones obtained in [5].
IV. CORRELATION FUNCTIONS AND SPECTRA OF QUANTUM STOKES
PARAMETERS
Since the Kerr medium is assumed to be lossless and dispersionless, the operatorsSˆ (t,z)
0
and Sˆ (t,z), as well as their dispersions, are conserved. Therefore, we focus our attention to
1
the quantum fluctuations of Sˆ (t,z) and Sˆ (t,z) by defining their correlation functions as
2 3
R (t ,t ) = Sˆ (t ,z)Sˆ (t ,z) − Sˆ (t ,z) Sˆ (t ,z) (k = 2,3). (14)
Sk 1 2 D k 1 k 2 E D k 1 ED k 2 E
The correlation functions (14) can be analytically computed by using Eqs. (5) and (7)-(8).
Here we simply write down the result for R (t,t+τ) in the approximation γ ,γ˜ ≪ 1:
S2 j
˜ ˜
R (t,t+τ) = δ(τ)+h(τ)[n¯ (t)φ (t)−n¯ (t)φ (t)]sin2[Φ (t)−Φ (t)]
S2 0,1 2 0,2 1 1 2
+ g(τ){n¯ (t)[φ2(t)+φ˜2(t)]+n¯ (t)[φ2(t)+φ˜2(t)]}sin2[Φ˜ (t)−Φ˜ (t)], (15)
0,1 2 2 0,2 1 1 1 2
where for simplicity we denote t = t, t = t + τ. In the absence of SPM [φ (t) = 0] and
1 2 j
˜
XPM [φ (t) = 0] of pulses, Eq. (15) gives us the correlation function for the coherent state
j
Rcoh(t,t+τ) = δ(τ), as expected. The spectral densities of quantum fluctuations of Sˆ (t,z)
S2 k
can be evaluated by using Wiener-Khintchine theorem: S (ω,t) = ∞ R (t,t+τ)eiωτdτ.
Sk −∞ Sk
Allowing for a small change of the envelope during the relaxation tiRme we obtain
S (Ω,t) = 1+2L(Ω)[n¯ (t)φ (t)−n¯ (t)φ (t)]sin2[Φ˜ (t)−Φ˜ (t)]
S2 0,1 2 0,2 1 1 2
+ 4L2(Ω){n¯ (t)[φ2(t)+φ˜2(t)]+n¯ (t)[φ2(t)+φ˜2(t)]}sin2[Φ˜ (t)−Φ˜ (t)], (16)
0,1 2 2 0,2 1 1 1 2
5
where L(Ω) = 1/(1 + Ω2), and Ω = ωτ is the reduced frequency. The spectral density
r
(16) depends on the relaxation time τ and quasi-statically changes with time. Besides, the
r
second term on r.h.s. of Eq. (16) indicates that the quantum fluctuations of Sˆ (t,z) can be
2
less than those corresponding to the coherent state Scoh(Ω,t) = 1. The correlation function
S2
of Sˆ (t,z), as well as its spectrum, can be easily obtained by shifting (15) and (16) in phase
3
with π/2.
For an arbitrary Ω = ω τ , at which the linear phase difference between incoming pulses
0 0 r
1 n¯ (t)φ (t)−n¯ (t)φ (t)
0,1 2 0,2 1
∆ϕ(t) = arctan 
opt 2 L(Ω ) n¯ (t)[φ2(t)+φ˜2(t)]+n¯ (t)[φ2(t)+φ˜2(t)]
 0 n 0,1 2 2 0,2 1 1 o
˜ ˜
+φ (t)−φ (t)−φ (t)+φ (t), (17)
1 2 1 2
is optimized, the expression (16) reaches the minimum value:
S (Ω ,t) = 1+2L2(Ω ){n¯ (t)[φ2(t)+φ˜2(t)]+n¯ (t)[φ2(t)+φ˜2(t)]}
S2 0 0 0,1 2 2 0,2 1 1
−2L(Ω ) [n¯ (t)φ (t)−n¯ (t)φ (t)]2
0 0,1 2 0,2 1
(cid:16)
+L2(Ω ){n¯ (t)[φ2(t)+φ˜2(t)]+n¯ (t)[φ2(t)+φ˜2(t)]}2 1/2. (18)
0 0,1 2 2 0,2 1 1
(cid:17)
To characterize the deviation of S (Ω,t) from the coherent level, we define the normal-
S2
∗
ized spectral variance S (Ω,t) = [S (Ω,t) − 1]/n¯ (t). In the case of the suppression of
S2 S2 0,1
∗ ∗
quantum fluctuations −1 ≤ S (Ω,t) < 0. Let us investigate the S (Ω,t) by choosing the
S2 S2
linear phase difference between pulses to be optimal at a defined reduced frequency Ω , and
0
change the intensity of one pulse (the control pulse) in comparison with the intensity of the
∗
otherone. Suchdependence ofS (Ω,t)att = 0andΩ = 0onthemaximumnonlinearphase
S2
addition φ ≡ φ (t = 0) in case the linear phase difference is optimal at Ω = 0 and Ω = 1,
0,1 1 0 0
is displayed in Figs. 1 and 2, respectively. Thus, at Ω = 0 the increase of the control pulse
0
intensity produces a uniform suppression of quantum fluctuation of Sˆ (t,z) for any φ > 1.
2 0,1
At Ω = 1 the increase of the control pulse intensity produces the suppression basically in
0
∗
the domain φ < 1. The normalized spectral variance S (Ω,t) at t = 0, Ω = 0, and
0,1 S2 0
ˆ
fixed φ = 2, is presented in Fig. 3. Now the suppression in S (t,z) is maximal at Ω ≈ 0.
0,1 2
Summarizing, the choice of the linear phase difference allows us to obtain the spectra with
the form of interest, and the increase of the control pulse intensity can effectively control
the suppression of quantum fluctuations of quantum Stokes parameters.
One can also control the suppression of quantum fluctuations of Sˆ (t,z) by increasing
2
the nonlinear coefficient γ in comparison with γ . This is equivalent to the increase of the
2 1
Kerr electronic nonlinearity for one pulse n in comparison with the one for another pulse
2(2)
∗
n (γ = β z). The variance S (Ω,t) at t = 0, Ω = 0 for various relations between
2(1) j j S2 0
nonlinear coefficients γ and γ is displayed in Fig. 4 in the simplest case of pulses with the
1 2
same intensity. In this case, the increase of γ incomparison with γ suppresses the quantum
2 1
fluctuations of Sˆ (t,z) basically at low frequencies, Ω ≈ 0 . A similar dependence, but at
2
Ω = 1, is shown in Fig. 5. Now, the squeezing takes place essentially at Ω ≈ 1 (ω ≈ 1/τ ).
0 r
Note the experimentally obtained squeezing of −2.8 dB in Sˆ reported in [10]. The
2
calculations in [10] use quantum noise operators and do not account for the finite relaxation
time of the Kerr nonlinearity. However, the relaxation time, which was accounted here, is
6
0
)
0
, -0.1
W(
2
S -0.2
*
S
-0.3

a
-0.4
b
c
-0.5
d
-0.6
0 0.5 1.0 1.5 2.0 2.5 3.0
f
0,1
∗
FIG. 1. Normalized spectral variance S (Ω,t) as a function of the maximum nonlinear phase
S2
addition φ at the reduced frequency Ω = 0 for initial phase difference ∆ϕ(t) which is optimal at
0,1
Ω = 0. Curves are calculated at time t = 0, γ = γ /4 = 2γ˜, and correspond to n¯ = n¯ /4 (a),
0 1 2 0,2 0,1
n¯ = n¯ /2 (b), n¯ = n¯ (c), n¯ = 3n¯ (d).
0,2 0,1 0,2 0,1 0,2 0,1

0
) d
0
, c
b
-0.1
W(
2 a
S
*
S
-0.2

-0.3
-0.4
0 0.5 1.0 1.5 2.0 2.5 3.0
f
0,1
FIG. 2. As in Fig. 1 but for Ω = 1.
0
7
0
)
0
, -0.1
W(
2
S
* -0.2
S
-0.3
a
-0.4
b c
-0.5 d
0 0.5 1.0 1.5 2.0 2.5 3.0
W
∗
FIG. 3. Normalized spectral variance S (Ω,t) at φ = 2 for initial phase difference ∆ϕ(t)
S2 0,1
which is optimal at Ω = 0. Curves are calculated at time t = 0, γ = γ /4 = 2γ˜, and correspond
0 1 2
to n¯ = n¯ /4 (a), n¯ = n¯ /2 (b), n¯ = n¯ (c), n¯ = 3n¯ (d).
0,2 0,1 0,2 0,1 0,2 0,1 0,2 0,1

0
)
0 a
-0.1
,
W( -0.2 b
2
S
*S -0.3
c
-0.4
-0.5 d
-0.6
e
f
-0.7
0 0.5 1.0 1.5 2.0 2.5 3.0
W
∗
FIG. 4. Normalized spectral variance S (Ω,t) at φ = 2 for initial phase difference ∆ϕ(t)
S2 0,1
which is optimal at Ω = 0. Curves are calculated at time t = 0, n¯ = n¯ , γ˜ = γ /2 and
0 0,1 0,2 1
correspond to γ = 2γ (a), γ = 3γ (b), γ = 4γ (c), γ = 5γ (d), γ = 6γ (e), γ = 7γ (f).
2 1 2 1 2 1 2 1 2 1 2 1
8
0
) a
0
,
-0.2
W( b
2
S
*
S
c
-0.4

d
-0.6 e f
-0.8
0 0.5 1.0 1.5 2.0 2.5 3.0
W
FIG. 5. As in Fig. 4 but for Ω = 1
0
of a fundamental importance since it determines the level of quantum fluctuations of Stokes
operators below the level corresponding to the coherent state [see Eq. (16)]. Besides, we
indicated the optimal strategy for the successful generation of the PS state in the electronic
Kerr medium.
V. CONCLUSION
We investigated the formation of polarization-squeezed light in a nonlinear medium with
electronic Kerr nonlinearity. The correlation functions and corresponding spectra of quan-
ˆ ˆ
tum Stokes parameters S and S were considered. We shown that, by adjusting the linear
2 3
phase difference between pulses, the maximum suppression of the quantum fluctuations of
Sˆ or Sˆ can be realized at the spectral component of interest. It is established that the
2 3
increase of the intensity of the control pulse can be employed to suppress the quantum fluc-
tuations of Sˆ . We find that the increase of one nonlinear coefficient (γ ) in comparison with
2 2
ˆ
another one (γ ) produces a substantial suppression of quantum fluctuations of S .
1 2
9
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