Intrinsic cross-Kerr nonlinearity in an optical cavity containing an interacting Bose-Einstein condensate

1.8 MB·

by  A. Dalafi

Intrinsic cross-Kerr nonlinearity in an optical cavity containing an interacting
Bose-Einstein condensate
A. Dalafi1 and M. H. Naderi2
∗
1 Laser and Plasma Research Institute, Shahid Beheshti University, Tehran 19839-69411, Iran
2Quantum Optics Group, Department of Physics, Faculty of Science,
University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran
(Dated: January 24, 2017)
An interacting cigar-shaped Bose-Einstein condensate (BEC) inside a driven optical cavity ex-
hibits an intrinsic cross-Kerr (CK) nonlinearity due to the interaction with the optical mode of
the cavity. Although the CK coupling is much weaker than those of the radiation pressure and
the atom-atom interactions, it can affect the bistability behavior of the system when the intensity
7
of the laser pump is strong enough. On the other hand, there is a competition between the CK
1
0 nonlinearityandtheatom-atominteractionsothatthelattercanneutralizetheeffectoftheformer.
2 Furthermore,theCKnonlinearitycausestheeffectivefrequencyoftheBogoliubovmodeoftheBEC
as well as the quantum fluctuations of the system to be increased by increasing the cavity driving
n
rate. However, in the dispersive interaction regime the effect of the CK nonlinearity is negligible.
a
Inaddition,weshowthatbyincreasingthes-wavescatteringfrequencyofatomiccollisionsonecan
J
generate a strong stationary quadraturesqueezing in the Bogoliubov mode of theBEC.
1
2
I. INTRODUCTION order of 1 where N is the number of atoms. There-
] √N
h fore, in many cases [16–20] one can neglect the CK term
p Hybrid systems consisting of Bose-Einstein conden- incomparisontotheradiationpressureinteraction. That
-
t sates (BEC)inside opticalcavities exhibit by themselves iswhywehavealsonotconsideredthe effectofthisterm
n optomechanical proprties [1–4]. The excitation of a col- onthedynamicsofthesysteminourpreviouspapers[21–
a
lective mode of BEC couples to the radiation pressure 24]. In addition to the radiation pressure and CK non-
u
q of the cavity optical field just like what happens for the linearities, there is another kind of nonlinearity which is
[ moving mirror in a bare optomechanical system [5–7]. duetotheatom-atominteractionintheBECwhichmay

One of the most important features of such systems is leadtosomeinterestingphenomenasuchasthequantum
1
their nonlinear proprties like the Kerr effect [8–10]. phase transition [25].
v
Recently, it has been proposed [11] and experimen- In this work, we are going to investigate under which
6
3 tally demonstrated [12] that a cross-Kerr (CK) type of conditions and in which regimes the effect of CK non-
0 couplingbetweenthemovingmirrorandtheopticalfield linearity manifest itself in a hybrid system consisting of
6 can be realized in a bare optomechanical cavity which a BEC traped inside an optical cavity. We show that it
0 is coupled to a Josephson junction. The CK nonlinear- canchangethe opticalbistability behaviorof the system
.
1 ity leads to a frequency shift in both the mechanicaland specially when the driving rate of the optical cavity is
0 opticalmodes[13] andalsoaffects the stability ofthe op- increased. Itisalsoshownthattheeffective frequencyof
7 tomechanical system [14]. In addition, it can change the the Bogoliubov mode of the BEC depends on the mean
1
optical bistability of the optomechanical system into a value of the optical mode through the CK coupling so
:
v tristable behavior [15]. that the Bogoliubov mode frequency gets larger by in-
i On the other hand, a hybrid system consisting of a creasing the number of intracavity photons. This is one
X
BEC traped inside anoptical cavityexhibits an intrinsic of the main results of our investigation.
r
a CK nonlinearity if the cavity is driven by a laser pump Moreover, we show that the CK effects get very small
whichis fardetuned fromthe atomic resonance. Inspite in the dispersive interaction regime where the effective
of bare optomechanical systems, in such hybrid systems cavity detunig is much larger than the damping rate of
thereisnoneedtouseanyaditionalcomponent(likethe the cavity. However, it increases the quantum fluctua-
Josephsonjunction)togeneratetheCKcouplingbecause tions near the bistability region and can also strengthen
here the CK nonlinearity emerges naturally due to the theatom-fieldentanglementinawiderangeofthecavity
directinteractionbetweentheatomsandtheopticalfield detuning. On the other hand, we investigate the non-
of the cavity. linear effect of atom-atom interaction against the CK
In fact, there exist two kinds of coupling between the nonlinearity and show how it neutralizes the CK effects.
Bogoliubov mode of the BEC and the optical field: the More importantly, it is shown that in the dispersive
radiation pressure and the CK coupling. However, the regime where the CK effect disappears, a steady-state
ratio of the CK to the optomechanical coupling is of the quadraturesqueezingintheBogoliubovmodeoftheBEC
canbe generatedbyincreasingthe s-wavescatteringfre-
quency of the atom-atom interaction.
The structure of the paper is as follows. In Sec. II we
∗ adalafi@yahoo.co.uk derivetheHamiltonianofthesystemconsistingofaBEC
2
inside an optical cavity, and in Sec. III the dynamics of
the system is described. In Sec. IV we study the CK
effect on the optical bistability, and in Sec.V we investi-
gate the CK effect on the entanglement and squeezing.
Finally, our conclusions are summarized in Sec. VI.
II. SYSTEM HAMILTONIAN FIG.1. (Coloronline)ABECtrappedinsideanopticalcavity
interacting with a single cavity mode. The cavity mode is
driven bya laser at rate η and thecavity decay rate is κ.
As depicted in Fig.1, the system we are going to in-
vestigate is an optical cavity with length L consisting of
a cigar-shapedBEC of N two-levelatoms with mass m
a
and transition frequency ω confined in a cylindrically quantum fluctuations of the atomic field about the clas-
a
symmetric trapwith a transversetrapping frequency ω sical condensate mode. By substituting the atomic field
andnegligiblelongitudinalconfinementalongthexdirec⊥- operator of Eq.(2) into Eq.(1), we can find the Hamilto-
tion. Anexternallaserwithfrequencyω andwavenum- nian of the system in the following form
p
ber k = ω /c drives the cavity at rate η = 2 κ/~ω
p p
P
through one of its mirrors (P is the laser powper and κ is H =~δca†a+i~η(a a†)+~Ωcc†c+~ζa†a(c+c†)
the cavity decay rate). −
1
If the laser pump is far detuned from the atomic res- + ~ωsw(c2+c†2)+~ga†ac†c. (3)
4
onance (∆ = ω ω exceeds the atomic linewidth
a p a
−
γ by orders of magnitude), the excited electronic state
of the atoms can be adiabatically eliminated and the Here, δ = ∆ + 1NU is the Stark-shifted cavity fre-
c − c 2 0
atomic spontaneous emission can be neglected [26–28]. quency due to the presence ofthe BEC,Ω =4ω +ω
c R sw
Inthisway,thedynamicsofthesystemcanbe described is the frequency of the Bogoliubov mode, ζ = √2NU
4 0
within an effective one-dimensional model by quantizing is the optomechanical coupling between the Bogoliubov
the atomic motional degree of freedom along just the x and the optical modes, ω = 8π~a N/m Lw2 is the s-
sw s a
axis. In the frame rotating at the pump frequency, the wave scattering frequency of the atomic collisions (w is
many-body Hamiltonian of the system reads thewaistradiusoftheopticalmode),andg = 1U isthe
2 0
CK coupling.
L/2 ~2 d2
H =Z L/2dxΨ†(x)h2−madx2 +~U0cos2(kx)a†a linAeasriitsysienenthferoHmamEiqlt.o(n3)iatnhoefrethaeresytshtreeme:ktihnedsraodfiantoionn-
−
1 pressure interaction between the Bogoliubov mode and
+ U Ψ (x)Ψ(x) Ψ(x) ~∆ a a+i~η(a a ),
2 s † i − c † − † the optical field (the fourth term), the nonlinear atom-
(1) atominteraction(the fifth term), andthe CK nonlinear-
ity (the sixth term). The important point is that the
where Ψ(x) and a are, respectively, the annihilation op- manifestation of the nonlinear CK term in the Hamil-
eratorsofthe atomic field andthe cavity mode. In addi- tonian of the system is due to the direct atom-photon
tion,∆ =ω ω isthecavity-pumpdetuningwhereω interaction. In other words, it is an intrinsic property of
c p 0 0
−
is the resonance frequency of the cavity. U = g2/∆ is the system.
0 0 a
the opticallatticebarrierheightperphotonwhichrepre-
Acomparisonbetweentheabove-mentionedthreenon-
sentstheatomicbackactiononthefield,g isthevacuum
0 linearterms showthatthe radiationpressureinteraction
Rabi frequency, Us = 4πm~2aas and as is the two-body s- is the most important term in the system because of the
wave scattering length [26, 27]. large value of the optomechanical coupling parameter ζ.
If the intracavity photon number is so low that the Thestrengthofthe atom-atominteractionisdetermined
icsonthdeitiroencoUil0fhrae†qauien≤cy10oωfRtheiscsoantdisefinesdatwehaetroemωs,Ra=nd2ℏumkn2a- bbyeitnhceresa-sweadvferosmcatzteerroinugpftroeqsuevenercaylωtesnwswofhωoRsebvyalmueanciapn-
der the Bogoliubov approximation [28], the atomic field ulating the transverse trapping frequency ω which can
operator can be expanded as the following single-mode change the waist radius of the optical mode⊥w [36].The
quantum field CK term whose strength is determined by the CK cou-
pling parameter g = U /2 has the weakest effect among
0
N 2 the others. In fact the ratio of the CK coupling to the
Ψ(x)= + cos(2kx)c, (2) optomechanical coupling is of the order of 1 . So, for
rL rL √N
very large values of N the CK term is negligible in com-
wherethefirsttermisthecondensatemodewhichiscon- parison to the radiation pressure and atom-atom inter-
sidered as a c-number and the operator c in the second actions. That is why in the previous works [21–24] this
term(theso-calledBogoliubovmode)correspondstothe term has been disregarded.
3
III. DYNAMICS OF THE SYSTEM drift matrix given by
κ ∆ G F
ThedynamicsofthesystemdescribedbytheHamilto- − I I
nian in Eq.(3) is fully characterized by the following set A= −∆ −κ −GR −FR , (8)
F F γ Ω( )
of nonlinear Heisenberg-Langevinequations  R I − − 
 GR GI Ω(+) γ 
− − − − 
a˙ = (iδ +κ)a iζa(c+c ) igac c η+√2κδa ,
− c − † − † − in where the new parameters in the drift matrix of Eq.(8)
i
c˙ = (iΩc+γ)c ωswc† iζa†a iga†ac+ 2γδcin. have been defined in terms of the real and imaginary
− − 2 − − p parts of the optical and atomic mean fields as follows
(4)
G =2α (ζ+gβ ), (9a)
R R R
Here γ characterizes the dissipation of the Bogoliubov G =2α (ζ+gβ ), (9b)
I I R
mode of the BEC. The optical field quantum vacuum
F =2gα β , (9c)
fluctuation δa (t) satisfies the Markovian correlation R R I
in
F =2gα β . (9d)
functions, i.e., hδain(t)δa†in(t′)i = (nph + 1)δ(t − t′), I I I
hδa†in(t)δain(t′)i = nphδ(t − t′) with the average ther- The solutionsto Eq.(7)arestable only if allthe eigen-
mal photon number nph which is nearly zero at optical values of the matrix A have negative real parts. The
frequencies [29]. Furthermore, δcin(t) is the quantum stability conditions can be obtained, for example, by us-
noise input for the Bogoliubov mode of the BEC which ing the Routh-Hurwitz criteria [31].
also satisfies the same Markovian correlation functions Basedonthedynamicalequations(7),thequadratures
as those of the optical noise [30]. The noise sources are of the Bogoliubov mode of the BEC oscillate effectively
assumeduncorrelatedforthedifferentmodesofboththe with the frequency ω =√Ω(+)Ω( ) which is given by
B −
matter and light fields.
In order to linearize the nonlinear set of equations (4) 1 3
one can expand the quantum operators around their re- ωB =r(cid:16)4ωR+ 2ωsw+g|α|2(cid:17)(cid:16)4ωR+ 2ωsw+g|α|2(cid:17).
spectiveclassicalmeanvaluesasa=α+δaandc=β+δc
(10)
where δa and δc are small quantum fluctuations around
Based on Eq.(10), the effective frequency of the Bo-
the mean fields α and β. In this way, a set of nonlin- goliubov mode depends not only on the s-wave scatter-
ear algebraicequations for the mean-field values and an-
ing frequency of the atom-atom interaction (ω ) but
sw
other set of linear ordinary differential equations for the also on the mean number of interacavity photons (α2)
quantumfluctuations willbe obtained. The steady-state | |
which is due to the presence of the CK nonlinearity. In
mean-field values are obtained as follows
the absence of the CK nonlinearity (g = 0), the effec-
η tive frequency of the Bogoliubov quadrature reduces to
α= , (5a)
the Bogoliubov frequency in the absence of the CK non-
−i∆+κ
linearity ω = ω which had been obtained in the
β = ζ α2 Ω(−)+iγ , (5b) previous wocrks [2B1|–g2=40].
− | | Ω(+)Ω( )+γ2
−
where∆=δ +2β ζ+g β 2withβ beingtherealpartof
thecomplexcmeanRfield|β,|andΩ( R) =Ω 1ω +g α2. IV. EFFECT OF THE CK NONLINEARITY ON
± c±2 sw | | OPTICAL BISTABILITY
Ontheotherhand,byintroducingtheopticalfieldand
the Bogoliubov mode quadratures as
InthissectionweshowhowtheCKnonlinearityaffects
the bistability behavior of the system. For this purpose
1 1
δX = (δa+δa†),δY = (δa δa†), (6a) we analyse our results based on the experimentally fea-
√2 √2i −
sible parameters given in [1, 2],i.e., we assume there are
1 1 N =105 Rbatomsinside anopticalcavityoflengthL=
δQ= (δc+δc†),δP = (δc δc†), (6b)
√2 √2i − 187µm with bare frequency ω = 2.41494 1015Hz cor-
0
×
responding to a wavelength of λ = 780nm. The atomic
the following linearized set of ordinary differential equa- D transitioncorrespondingtothe atomictransitionfre-
2
tions is obtained for the quantum fluctuations quency ω =2.41419 1015Hz couples to the mentioned
a
×
mode of the cavity. The atom-field coupling strength
δu˙(t)=Aδu(t)+δn(t), (7) g =2π 14.1MHzandtherecoilfrequencyoftheatoms
0
×
isω =23.7KHz. Furthermore,weassumethattheequi-
R
where δu(t) = [δX,δY,δQ,δP]T is the vector of con- libriumtemperatureofthe BEC isT =0.1µK.Forthese
tinuous variable fluctuation operators and δn(t) = set of experimental data the the CK coupling parameter
[√2κδX ,√2κδY ,√2γδQ ,√2γδP ]T is the corre- is three orders of magnitude smaller than the optome-
in in in in
sponding vector of noises. The 4 4 matrix A is the chanical coupling parameter (g 0.004ζ).
× ≈
4
1.0 4
(a) = , sw= R (a)η=2κ,ωsw=5ωR
0.8
3
withCK
0.6
2 2 nonlinearity
α α 2 withoutCK
0.4 |
nonlinearity
0.2 1
0.0
0
-10 -5 0 5 10
-10 -5 0 5 10
δ κ
c δc/κ
4
4
(b)η=2κ,ωsw=ωR
(b)η=2κ,ωsw=10ωR
3
3
withCK
2 nonlinearity
α| 2 2α 2
wihout |
1 CK
1
nonlinearity
0
0
-10 -5 0 5 10
-10 -5 0 5 10
δc/κ δc/κ
FIG. 2. (Color online) The mean number of intracavity
FIG. 3. (Color online) The mean number of intracavity
photonsversusthenormalizeddetuningδc/κfortwovaluesof photons versus the normalized detuning δc/κ for two values
theexternallaserdrivingrate: (a)η=κand(b)η=2κwhen
of the s-wave scattering frequency: (a) ωsw = 5ωR and (b)
ωsw = ωR. The red (blue) line corresponds to the absence ωsw = 10ωR when the pump rate is η = 2κ. The red (blue)
(presence) of the CK nonlinearity. The other parameters are
line corresponds to the absence (presence) of the CK nonlin-
L = 187µm, λ = 780nm and κ = 2π×1.3MHz. The cavity
earity. The other parameters are the same as those in Fig.2.
containsN =105 Rbatomswith ωsw =ωR andγc =0.001κ.
ing the s-wave scattering frequency from ω = ω in
sw R
InFig.2wehaveplottedthemeannumberofintracav- Fig.2(b) to ω = 5ω in Fig.3(a) the two curves get
sw R
ity photons versus the normalized detuning δc/κ for two nearer to each other. Finally, by increasing the s-wave
values of the external laser driving rate η = κ [Fig.2(a)] scattering frequency up to ω = 10ω in Fig.3(b) the
sw R
and η = 2κ [Fig.2(b)] when the s-wave scattering fre- CK effect is completely neutralized by the atom-atom
quency has been fixed at ωsw =ωR. The red (blue) line interaction and therefore the two curves overlap com-
corresponds to the absence (presence) of the CK non- pletely.
linearity. As is seen from Fig.2(a) for a driving rate of Intheprevioussection,itwasshownhowthepresence
η = κ the two curves (red and blue lines) overlap with of the CK nonlinearity causes the effective frequency of
each other completely. It means that for small driving the Bogoliubovmode (ω )tobe dependentonthemean
B
rates the presence of the CK nonlinearity cannot affect number of interacavity photons [Eq.(10)]. Due to the
the bistability of the system. However,by increasingthe dependence of α2 on the effective cavity detuning δ
c
intensity of the pump laser at a fixed value of ωsw the (as depicted in|F|igs.2 and 3), the effective Bogoliubov
effect of the CK nonlinearity manifest itself specifically frequency also depends on δ . In Fig.4 the normalized
c
in the bistabilty region. This can be seen in Fig.2(b) effectiveBogoliubovfrequency(ω /ω )hasbeenplotted
B c
where the twocurves areresolvedfromeachother in the versusthenormalizeddetuningδ /κfortheexternallaser
c
bistability region. pump rate η =2κ and ω =ω . As is seen from Fig.4,
sw R
On the other hand, the nonlinear effect of atom-atom far away from the bistability region where δc κ the
| | ≫
interaction can neutralize the effect of CK nonlinear- effective Bogoliubovfrequency (ωB) gets nearto the Bo-
ity. In order to illustrate this procedure, in Fig.3 the goliubovfrequencyintheabsenceoftheCKnonlinearity
mean number of intracavity photons has been plotted (ωc)whilenearthebistabilityregionwhere3κ<δc <9κ
versus the normalized detuning δ /κ for two values of the effectiveBogoliubovfrequencyhasthe maximumde-
c
ωsw = 5ωR [Fig.3(a)] and ωsw = 10ωR [Fig.3(b)] when viation from ωc because in this region the mean field of
the pump rate has been fixed at η =2κ. The red (blue) the optical mode has the maximum value as has been
linecorrespondstotheabsence(presence)oftheCKnon- shown in Fig.2(b).
linearity. As is seen, for a fixed pump rate by increas- In order to examine the bistability behavior in terms
5
5
(a)
1.15 withCK
4
nonlinearity
ωc 1.10 3
/B 2
ω α
| 2
1.05 withoutCK
1 nonlinearity
1.00
-10 -5 0 5 10 0
0 2 4 6 8 10
δc/κ
ηκ
FIG. 4. (Color online) The normalized effective Bogoliubov 1.20
frequency (ωB/ωc) versus the normalized detuning δc/κ for (b)
η=2κandωsw =ωR. Theotherparametersarethesameas 1.15
those in Fig.2.
c
ω
/B 1.10
ω
of the pump laser rate, in Fig.5(a) we have plotted the
1.05
mean number of intracavity photons versus the normal-
izedpumplaserrateη/κforδ =5κandω =ω . The
c sw R 1.00
red (blue) line corresponds to the absence (presence) of
0 2 4 6 8 10
the CK nonlinearity. As is seen, for small values of the
η/κ
pump laser rate when the system is in the lower branch
the two curves overlap while for large pump rates when
FIG. 5. (Color online) (a) The mean number of intracavity
the system is in the upper branch the two curves are
photonsversusthenormalizedpumplaserrateη/κ. Thered
resolved from each other. It means that for a specified (blue) line corresponds to the absence (presence) of the CK
detuning the effect of CK nonlinearity manifests itself nonlinearity. (b) The normalized effective frequency of the
when the driving laser becomes more intensive. That is Bogoliubov quadrature(ωB/ωc)versusthenormalized pump
also why in Fig.2(a) the two curves overlap. laser rate η/κ. Here, δc = 5κ,ωsw = ωR and the other pa-
In Fig.5(b) the normalized effective Bogoliubov fre- rameters are thesame as those in Fig.2.
quency (ω /ω ) has been plotted versus the normal-
B c
ized laser pump rate η/κ for detuning δ = 5κ and
c Langevinequations[Eq.(7)], one canshowthatV fulfills
ω = ω . As is seen, for η < 1.5κ the CK nonlinear-
sw R the Lyapunov equation [32]
ity does not change the effective Bogoliubov frequency
so that ωB ωc (corresponding to the lower branch AV +VAT = D, (11)
≈ −
in Fig.5(a)). At the threshold of the bistability where
where
η 1.5κtheeffectiveBogoliubovfrequencymakesasud-
≈
denjumpanditsvalueincreasesby10percentduetothe
D =Diag[κ,κ,γ(2n +1),γ(2n +1)], (12)
c c
effect of the CK nonlinearity. Beyond the threshold, the
deviation of the effective Bogoliubov frequency from ωc is the diffusion matrix with nc =[exp(~ωB/kBT −1)]−1
increases by increasing the pump rate. asthemeannumberofthermalexcitationsoftheBogoli-
ubovmodeinthepresenceoftheCKnonlinearity(inthe
absence of CK, ω should be substituted for ω ). The
B c
V. EFFECT OF THE CK NONLINEARITY ON Lyapunov equation (11) is linear in V and can straight-
ENTANGLEMENT AND SQUEEZING forwardly be solved. By solving Eq.(11) we can obtain
thematrixV whichgivesusthesecond-ordercorrelations
of the fluctuations.
In the previous section we studied the effect of the
Atfirst,weexaminetheatom-fieldentanglementwhich
CK nonlinearity on the mean-field values of the system.
is calculated by the logarithmic negativity[33]:
In this section we will show how the quantum fluctu-
ations and correlations of the system may be affected E =max[0, ln2η ], (13)
N −
by the presence of the CK nonlinearity when the sys- −
1/2
tem reaches to the stationary state. Due to the lin- where η 2 1/2 Σ(V) Σ(V)2 4detV is the
− −
earized dynamics of the fluctuations and since all noises lowest sym≡plectic ehigenval−uepof the p−artial trianspose of
are Gaussian the steady state is a zero-mean Gaussian
the matrix V which can be written as
state which is fully characterized by the 4 4 station-
×
ary correlation matrix (CM) V, with components V =
ij V = A C , (14)
δui( )δuj( )+δuj( )δui( ) /2. Usingthequantum (cid:18) T (cid:19)
h ∞ ∞ ∞ ∞ i C B
6
20
0(cid:0)00(cid:1)0 withCKnonlinearity (a)
0.0028 15 withoutCK
withCKnonlinearity
withoutCK
> nonlinearity
EN 0.0026 nonlinearity cδ 10
†
c
δ
0.0024 <
5
0.0022
-4 -2 0 2 4 6 8 10 0
δc/κ -4 -2 0 2 4 6 8 10
δc/κ
20
FIG. 6. (Color online) Atom-field entanglement versus the
(b)
normalized detuningδc/k for η=7κandωsw =ωR. Thered
(blue) line corresponds to the absence (presence) of the CK 15 withCKnonlinearity
nonlinearity. The other parameters are the same as those in
Fig.2.
Q 10
S
withoutCK
nonlinearity
and Σ(V)=det +det 2det . 5
A B− C
In Fig.6 the atom-field entanglement has been plotted
versusthenormalizeddetuningδ /κinthepresence(blue 0
c
line)andintheabsence(redline)oftheCKnonlinearity -4 -2 0 2 4 6 8 10
for the pumping rate η = 7κ and the s-wave scattering δc/κ
frequency ω =ω , in the the rage of δ where the sys-
sw R c
tem is stable. As is seen, the CK nonlinearity increases FIG.7. (Color online) (a) theincoherent excitation number
of atoms in the Bogoliubov mode and (b) the squeezing pa-
the atom-field entanglement to some extent for a wide
rameterofthequadratureδQversusthenormalizeddetuning
range of the effective detuning δ . It is similar to what
happens in the bare optomechancical systems where the δc/κ. The red (blue) line corresponds to the absence (pres-
ence) of the CK nonlinearity. Here, η = 2κ,ωsw = ωR and
presence of the CK nonlinearity strengthen the entan-
theother parameters are thesame as those in Fig.2.
glement between the moving mirror and the optical field
as has been recently shown in Ref.[35]. In the present
hybrid system the effect of the CK nonlinearity on the tum fluctuations and also on the squeezing behavior of
atom-field entanglement is very weak for small pumping the system.
rates. Asiswellknown,forasingle-modequantumfieldwith
On the other hand, the nonlinear atom-atom interac- quadratures q and p obeying the commutation relation
tion, i.e., the fifth term in Eq.(3), is analogous to the [q,p] = i, the degree of squeezing is defined in terms of
interaction Hamiltonian of a degenerate parametric am- the squeezing parameters S = 2 (∆q)2 1 and S =
q p
h i−
plifier(DPA). InaDPAapumpbeamgeneratesasignal 2 (∆p)2 1 where (∆q)2 = q2 q 2 and (∆p)2 =
beam by interacting with a χ(2) nonlinearity. This pro- ph2 ip−2 are thehquantuim huncie−rtahinities. hWheneiver
h i − h i
cess has long been considered as an important source of S < 0(i = p,q), the corresponding state is a squeezed
i
thesqueezedstateoftheradiationfield[34]. Ashasbeen one.
shown in Ref.[22], an interacting BEC behaves as a so- Based on the experimental data of Refs.[1, 2], our nu-
called”atomicparametricamplifier”(APA)inwhichthe mericalresultsshowthatthestationarysqueezingispos-
condensate acts asanatomic pump field andthe Bogoli- sible just for the quadrature δQ of the Bogoliubovmode
ubovmodeplaystheroleofthesignalmodeintheDPA. whosesqueezingparameterS iscalculatedfromthefol-
Q
Also, the s-wave scattering frequency of atom-atom in- lowing equation
teraction plays the role of the nonlinear gain parameter.
S =2 δQ2 1=2V 1. (15)
In the very simplified model studied in Ref.[22] where Q 33
h i− −
the cavity is not driven continuously by a pump laser, it
Furthermore,the incoherentexcitation number of atoms
was shown how a squeezed state can be generated dis-
in the Bogoliubov mode is calulated as
regarding all damping processes. Here, we would like
to show that in a realistic model where the cavity is 1
driven by an external pump laser and in the presence δc†δc = (V33+V44 1). (16)
2 −
ofalldampingprocesses,onecangenerateasteady-state (cid:10) (cid:11)
quadraturesqueezingintheBogoliubovmodeoftheBEC The incoherent excitation number of atoms in the
throughthe interatomic interactions. In addition, we in- Bogoliubov mode and the squeezing parameter of the
vestigate the effect of the CK nonlinearity on the quan- quadrature δQ have been, respectively, plotted in
7
0.00 to S < 0.3.
Q
−
-0.05 Although the nonlinear atom-atom interaction leads
to squeezing of the matter field of the BEC, it cannot
-0.10
causethe quadraturesqueezingofthe opticalfield ofthe
Q -0.15 cavity. However,onecancontrolthe squeezingofthe Bo-
S
-0.20 goliubov mode of the BEC by the s-wave scattering fre-
quency of atom-atom interaction which itself is control-
-0.25
lable through the transversetrapping frequency ω [36].
- Since the quadrature squeezing occurs in the disp⊥ersive
(cid:2)(cid:3)(cid:4)(cid:2)
0 10 20 30 40 50 regime, it is not affected by the CK nonlinearity which
manifests itself jut in the nondispersive regime.
ωsw/ωR
FIG. 8. (Color online) The squeezing parameter SQ of the
quadrature δQ of the Bogoliubov mode of the BEC versus VI. CONCLUSIONS
the normalized s-wave scattering frequency ωsw/ωR for δc =
−15κ and η = 5κ. The other parameters are the same as
those in Fig.2. In conclusion, we have studied a driven optical cavity
containingacigar-shapedBEC.Intheweaklyinteracting
regime, where just the first two symmetric momentum
Figs.7(a) and 7(b) versus the normalized detuning δ /κ sidemodes areexcitedbythe fluctuations resultingfrom
c
for η =2κ and ω =ω in the presence (blue line) and theatom-lightinteraction,theBECcanbeconsideredas
sw R
in the absence (red line) of the CK nonlinearity. As is a single mode quantun field in the Bogoliubov approxi-
seen, the quantum fluctuations in the excitation number mation. In this way, the Bogoliubov mode of the BEC
of the Bogoliubov mode as well as the squeezing param- is coupled to the optical field in two ways: the radiation
eterS divergenearthe bistabilityregion. Thepresence pressure interaction and the CK coupling.
Q
of the CK nonlinearity increases the slope of the curve Although the CK coupling is much weaker than those
and make the fluctuations diverge more rapidly. How- oftheradiationpressureandtheatom-atominteractions,
ever, in the dispersive regime where δc κ the quan- it manifests its effect when the intracavity optical field
| | ≫
tum fluctuations fade away so that δc†δc as well as SQ is strong enough (when the driving rate of the cavity is
h i
tend to zero both in the presence and in the absence of increased). ThemostimportantoutcomeoftheCKnon-
the CK nonlinearity. linearity is that the Bogoliubov frequency of the BEC is
Therefore, the effects of the CK nonlinearity are ap- dependent on the mean value of the optical field which
peared in the nondispersive regime where δc κ while leads to a change of bistability behavior. On the other
| | ≈
in the dispersive regime neither the mean fields nor the hand, the nonlinear effect of atom-atominteractionneu-
quantum fluctuations are affected by the CK nonlinear- tralizes the CK effect when the s-wave scattering fre-
ity. Instead, the nonlinear atom-atom interaction can quency is increased.
change the properties of the system in both regimes. Furthermore,theCKnonlinearitycausesthequantum
Toverifythisresultmoreclarly,inFig.8,wehaveplot- fluctuations of the system to be increased more rapidly
ted the squeezing parameterSQ of the quadratureδQ of near the bistability region. However, in the dispersive
the Bogoliubov mode of the BEC versus the normalized regime where the CK effect disappears, a steady-state
s-wave scattering frequency ωsw/ωR for δc = 15κ and quadraturesqueezingintheBogoliubovmodeoftheBEC
−
η = 5κ. As can be seen, the blue and red lines corre- canbe generatedbyincreasingthe s-wavescatteringfre-
sponding to the presence and absence of the CK nonlin- quency of the atom-atom interaction.
earity have a complete overlap because the system is in
thedispersiveregime. Intherangeofthosevaluesofω
sw
considered in Fig.8 the system is stable and the Bogoli-
ACKNOWLEDGEMENT
ubovapproximationaswellasthe singlemode condition
oftheBECisfulfiled. Asisseen,the degreeofsqueezing
isincreasedbyincreasingthes-wavescatteringfrequency A.D wishes to thank the Laser and Plasma Research
so that for ω > 30ω the degree of squeezing reduces Institute of Shahid Beheshti University for its support.
sw R
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