Quantum correlations of two optical fields close to electromagnetically induced transparency

0.21 MB·English

by  Alice Sinatra

Quantum correlations of two optical fields close to electromagnetically induced
transparency
A. Sinatra
Laboratoire Kastler Brossel, ENS, 24 Rue Lhomond, 75231 Paris Cedex 05, France
We show that three-level atoms excited by two cavity modes in a Λ configuration close to elec-
tromagnetically induced transparency can produce strongly squeezed bright beams or correlated
beams which can be used for quantum non demolition measurements. The input intensity is the
experimental “knob” for tuning the system into a squeezer or a quantum non demolition device.
The quantum correlations become ideal at a critical point characterized by the appearance of a
switching behavior in the mean fields intensities. Our predictions, based on a realistic fully quan-
tum 3-level model including cavity losses and spontaneous emission, allow direct comparison with
futureexperiments.
7
0
PACSnumbers: 42.50.Dv,42.50.Gy,42.65.Pc
0
2
2e
n Usingopticalnonlinearitiesitispossibletomanipulate 3
a
optical beams to the level of quantum fluctuations, pro-
J
ducing squeezed states [1] that are important resources
8 a a
for quantum information with continuous variables [2]. 1 2
1
Related to the field of squeezing is that of quantum

3 nondemolition(QND)measurementsonanopticalfield,
2
v wherequantumcorrelationsbetweentwodifferentmodes 1
6
oftheelectromagneticfieldareexploitedtoovercomethe
1
back-actionnoiseofaquantummeasurement[3]. Besides
0 FIG. 1: Two cavity modes interact with the atoms in a Λ
3 thefundamentalinterestinthetheoryofmeasurement,it configuration close to EIT conditions.
0 was shown that QND correlations of propagating beams
6 have direct application in quantum communication pro-
0
tocols as teleportation [4]. The best single back-action-
h/ evading measurement on optical beams was performed atomictransition. Wedefine∆j = ωajγ−wωj theatomicde-
tuningsnormalizedtothedecayrateoftheopticalcoher-
p using coldatoms inside a doubly resonantcavity[5]. We
- suggest that these performances could be significantly encesγw =(γ1+γ2)/2whereγ1+γ2 isthe totalpopula-
nt improved by tuning the system close to the electromag- tiondecayrateoftheupperlevel;θj = ωcjκ−jωj thecavity
a detunings normalized to the cavity decay rates κ , and
netically induced transparency (EIT) conditions [6]. j
u
g2N
q Alreadyinthenineties,theoreticalstudiesshowedthat Cj = γwjκj the cooperativities where gj are the coupling
: constantsforthetwoconsideredtransitions. Weusenor-
v a lambda three-level medium close to EIT conditions in
i malized variables proportionalto the intracavityand in-
X a cavity can be used to obtain squeezing [7]. Contrarily
r to previous proposals, here we assume that two different put fields xj = √γ2wgjhaji and yj = √γ2wgj √2TjEjin respec-
a modes are resonant in the cavity. For small and sym-
tively,whereT isthe(input-output)mirrortransmissiv-
j
metrical detunings from the upper level of Λ three-level
ityforthefieldj. Wenamevandwthenormalizedpolar-
atoms (see Fig.1), absorption is suppressed and the dis- izations between levels 1-3 and 2-3 v = (√2/N) R ,
−
persive non linear response gives rise to a rich scenario w = (√2/N) S where R and S are−collectivehoperi-
−
where either self correlations (squeezing) or cross QND − h i
ators constructed from the single atom operators 1 3
correlationscanbeestablishedintheoutputbeams. The | ih |
and 2 3 as in [9]. The master equation and the semi-
correlationsbecomeidealatacriticalpointthatwechar- | ih |
classicalequationsdescribingtheΛsystemwithtwocav-
acterizeanalytically. The technique weproposeisexper-
ity fields, with the same notations introduced here, are
imentally accessible, and first experimental steps in this
givenanddiscussedindetailin[10]wherethismodelwas
directions were done in atomic vapors without a cavity
successful to reproduce the experimental results of [5].
[8]. Hereweshowthatthepresenceofthecavityisacru-
Let us consider a set of parameters symmetric for the
cialadvantageespeciallyifonecanreachthegoodcavity
two transitions: y = y , C = C, γ = γ, κ = κ,
limit. | j| | | j j j
θ = 0 (empty cavity resonance for both fields), and let
j
WeconsiderN atomsinacavityexcitedbytwomodes ∆ = ∆ =ǫ be small and positive. In Fig. 2 we show
1 2
−
symmetricallydetunedfromtheupperlevelofaΛthree- in rescaled units the stationary intensities of the intra-
level scheme as in Fig.1. For j = 1,2 let ω be the fre- cavity fields I = x 2/4Cǫ as a function of the common
j j j
| |
quencyoffieldjandh¯ω theenergyofthecorresponding intensity of the input fields Y = y 2/4Cǫ. With a solid
aj
| |
2
2.0 1.2 1.2
I = I 1.0 1.0
1.5 1 2 0.8 0.8
0.6 0.6
I , I1201..50 I1 I I = I I100..24 I200..24
2 1 2 0.0-3 -2 -1 0 1 2 3 0.0-3 -2 -1 0 1 2 3
0.0 q q
1 1
0.0 0.5 1.0 1.5 2.0
1.2 1.2
Y = Y
1 2 1.0 1.0
0.8 0.8
FIG.2: StationaryintensitiesoftheintracavityfieldsI1,and
0.6 0.6
I2 as a function of the common intensity of the input fields I1 I1
Y1 = Y2 = Y. In solid line the solution I1 = I2. The thick 0.4 0.4
(red thin) line correspond to stable (unstable) solutions. In 0.2 0.2
dashed-dotted line one of the two stable solutions with I1 6= 0.0 0.0
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
I2. Parameters: 2ǫ = 0.125 and C1 = C2 = 250, γ1 = γ2 = q q
10κ1, κ2 =κ1, θ1=θ2=0. 1 1
FIG. 3: Intracavity field intensities I1 (left half), I2 (right
half) across the cavity scan. Upper half: Y = 1.05. Lower
line we have plotted an S-shaped solution with I1 = I2. half: Y =0.95. Thethick(redthin)linescorrespondtostable
A stable branch of this solution appears for Y >1. The (unstable) solutions. The otherparameters are as in Fig. 2.
negativeslope branchandthe lowerbranchveryclose to
zero intensity are both instable and play no role in the
following. For Y < 1, apart from the solution I1 = I2, ia†jeiφ, the squeezing spectrum is defined as
we get two other solutions with I = I . In the figure
1 2
we show one of them with I > I .6 The second one is Sφ(ω)=1+2κ ∞ e iωt :δXφ(t)δXφ(0): dt (1)
1 2 j j − h j j i
obtained by exchanging I1 and I2. Both solutions are Z−∞
stable in the considered case θ1 =θ2 =0. where δXφ denotes the time dependent fluctuation of
j
We choose now two values of the input intensity, in
the operator Xφ around a steady state point. The col-
turn above and below the turning point Y = 1, and j
umn indicates normal and time ordering for the product
show the stationarysolutions for intracavityfields inten-
inside the mean. Sφ = 1 is the shot noise and Sφ = 0
sities as the cavity detunings vary in Fig. 3. The sta- j j
meanstotalsuppressionoffluctuationsinthequadrature
ble branches of these curves (thick lines) can be easily
Xφ. The crossed correlations between the two fields are
obtained experimentally by sweeping the cavity length j
described by the coefficients C , C and V [11] char-
[10]. Wevaryθ1 andθ2 keepingthemalwaysequalwhich acterizing a QND measure of tshe ammplitudse|mquadrature
would imply the use of two driving fields of close optical
Xin ofonefield,thesignal, performingadirectmeasure-
frequencies ∆λ/λ 1 (and for example different polar-
≪ ment on the phase quadrature Yout of the other field,
izations). For Y = 1.05 i.e. 5% above the turning point
the meter. Among the three coefficients C quantifies
(upper half of Fig. 3) the stable solutions for the intra- s
the non-destructive character of the measurement, C
cavity intensities are Lorentzian-looking curves symmet- m
itsaccuracyandV referstothetothe“quantumstate
ricallyshiftedbyasmallamountfromtheirempty-cavity preparation” capasb|imlities of the system.
positions for both fields. Only for θ = θ = 0 the two
1 2
fields have the same stationary amplitude in the cavity C = C(Xin,Xout), C =C(Xin,Yout), (2)
s m
corresponding to the stable high-transmission branch of
V = Xout,Xout 1 C(Xout,Yout) (3)
the S-shaped curve in Fig.2. For Y =0.95 i.e. 5% below s|m h i −
the turning point (lower half of Fig. 3) the situation is where for two operators A(cid:0)and B we define (cid:1)
rather different: the stable solution for the two fields
switches between a high-intensity and a low-intensity A,B 2
C(A,B)= |h i| with (4)
curve being always I = I although y = y . In con- A,A B,B
1 2 1 2
6 | | | | h ih i
trastwiththepreviouscasethissituationisveryfarfrom
tfahcetisntdreopnegnlydecnotu-pfileeldd.s EIT solution and the fields are in A,B = +∞e−iωt1 A(t)B+BA(t) dt. (5)
h i 2h i
Let us now introduce the usefull correlationsto carac- Z−∞
terize the quantum fluctuation properties of the system. Superscripts in and out refer to the input and output
For a givenquadrature of the jth field: Xφ = a e iφ+ cavity fields. By calling φin and φout the phases of the
j j − j j
3
input and output fields in steady state, and choosing 1.0
field 1 as the meter and field 2 as the signal, we define
0.8
XQNouDt(imn)e=asuXr2eφm2ouet(nint)Cand=YCout==1Y,1φao1nudt+Vπ/2. =Fo0r.an ideal zing 0.6
m s sm e
Thequantumfluctuationscounterparto|fFig.3(top)is ue 0.4
q
showninFig.4(top)wheresqueezingoftheoutputfields S 0.2
optimizedwith respectto the quadratureSbest(ω =0)is
j 0.0
plotted as a function of the cavity detuning. A large 0.0 0.5 1.0 1.5 2.0
amount of squeezing is present in both fields close to w /k
θ1 =0. AsonecanseefromFig.3(top)thetwofieldsare 1.0
welltransmittedbythecavityforθ1 =0,andthesystem nts 0.8
efficiently converts the input coherent beams into bright e
ci
squeezedbeams. Correspondinglyto Fig. 3 (bottom) for effi 0.6
Y = 0.95, in Fig.4 (bottom) we plot the coefficients Cs, o 0.4
C
Cm and Vsm across the cavity scan. The useful quan- D 0.2
|
tumcorrelationsarecalculatedbyalinearizedtreatment N
ofquantumfluctuations aroundthe stable stationaryso- Q 0.0
0.0 0.2 0.4 0.6 0.8 1.0
lutionasin[10]. Despitethefactthatthetwofieldshave
w /k
different intracavity intensities at θ =0, they play here
1
symmetrical roles for the QND scheme; the figure corre-
FIG. 5: Top: Squeezing spectra of field 1, for Y = 1.05 in
sponding to the reversed scheme 1 2 being obtained
↔ thecenterofthecavityscan(θ1 =0.0013). Bestsqueezingin
by reflection of the plots θ1 ↔−θ1. solidlineandamplitudesqueezingS1φo1ut inpurpledashedline.
Bottom: QNDspectraforY =0.95andθ1 =0.0018. Cm(red
1.0 dashed-dotted line), Cs (solid line), Vs|m (blue dashed line).
g The otherparameters are as in Fig.2.
n 0.8
zi
e 0.6
e
u in this regime, representing a significant improvement
q 0.4
s with respect to previously obtained results [5] based on
est 0.2 the so called “ghost transition” scheme [12], [10]. Al-
B
0.0 though we concentrate here on the good cavity limit, in
-1.0 -0.5 0.0 0.5 1.0
which as we will show the quantum correlations become
q
ideal approaching the turining point Y = 1, some QND
1
1.0 correlations between the two modes persist also in the
nts 0.8 bad cavity limit. For example for ǫ = 0.25, C = 25,
e κ = 3γ, θ = 6 10 3, Y = 0.9 and ω = 0.1γ we get
oeffici 00..46 CsIn=tChemli=m1 i0t.7o2f,w×Vesa|km−a=to0m.2ic6.detunings,usefulanalytical
c
D 0.2 results can be obtained. The analytical solution of the
N semiclassical equations of the system at steady state is
Q 0.0
givenin[10]. Byexpandingthesteadystatepolarizations
-1.0 -0.5 0.0 0.5 1.0
v and w to the first order in ǫ we obtain
q
1
4ǫx x 2 4ǫx x 2
1 2 2 1
v =i | | w = i | | . (6)
FIG. 4: Top: Best squeezing of the fields across the cavity (x 2+ x 2)2 − (x 2+ x 2)2
1 2 1 2
| | | | | | | |
scan for Y = 1.05 and ω = 0. Squeezing of field 1 (2) is
plotted withasolid (dashed)line. Bottom: QNDcoefficients Byinserting(6)intheequationsfortheintracavityfields
across the cavity scan for Y = 0.95 and ω = 0. Cm (red amplitudes, with yj = y , θj = 0 and Cj = C, we
| | | |
dashed-dotted line), Cs (solid line), Vs|m (blue dashed line). obtainat steady state a “universalsolution” for rescaled
Parameters as in Fig.3. fieldintensities. ForY <1therearetwostablesolutions
Y Y
We show in Fig. 5 the frequency dependence of the I1 = (1 η) ; I2 = (1 η) (7)
2 ± 2 ∓
quantum correlations both below and above the turning
pointY =1,forafixedvalueofthecavitydetuningclose where η =√1 Y2. For Y >1, out of two solutions
−
to zero. For values of the cooperativity parameters cur-
Y 1
rentlyobtainedinexperiments,QNDcoefficientssuchas
I =I =I ; I = 1 1 (8)
Cs = 0.98, Cm = 0.95, Vsm = 0.05 can be acheived 1 2 2 ±r − Y2!
|
4
the one with the plus sign is stable and the other one curve(Fig.2)whichdividestheparameterspaceintotwo
unstable. Solutions (7)-(8) are indistinguishable from parts: for input intensities higher than the upper turn-
those of the full three-level model in Fig. 2. The ing point of the curve, the quantum fluctuations of the
phases of the input fields with respect to intracavity fields become quadrature dependent and can be reduced
fields (which are taken real at steady state) are φin = in a quadrature, while for input intensities lower than
1
atan I /I , φin = atan I /I for Y < 1, and the turning point, crossed phase-intensity quantum cor-
2 1 2 − 1 2 relations build up between the two fields. The system
φi1n =hpatan[1i/2I] = −φi2n forhpY > 1i. For the output becomes a perfect “squeezer” or an ideal QND device at
fields φout = φin, φout = φin in both cases.
1 − 1 2 − 2 the turning point. The “universal” point Y = 1, can
Inordertostudythequantumpropertiesofthesystem
be identified experimentally by the appearance of the
analyticallywefurtherassumethat(i)γ κsothatthe
≫ switching behavior described in Fig.3, and can be used
atomic fluctuations follow adiabatically the field fluctu-
asareferenceintheparameterspacetochooseeitherthe
ations, and (ii) the noise from spontaneous emission is
squeezing or the QND effect and to optimize it. An im-
negligible, which we found true when the cooperativity
plementation using either a vapor [8], or a trapped cold
is large enough. In this limit, using the steady state po-
atoms in an optical cavity [5],[14],[15] seems within the
larizations (6), we can solve analytically the equations
reach of present technology.
for the fieldfluctuations andobtainthe correlationfunc-
tions.
For Y >1 and I =I =I and taking κ 1 as the unit I thank L. Lugiato and P. Grangier for useful discus-
1 2 −
of time, we obtain sions, M. Guerzoni for her contribution to this work and
Y.Castinforcommentsonthemanuscript. LKBisUMR
( 1)3−j 8552 du CNRS de l’ENS et de l’UPMC; support from
δx˙j =−δxj +i −2I δx∗j j =1,2. (9) IFRAF is acknowledged.
These equations describe two independent two-photon
processesforwhichinstabilities andsqueezinghavebeen
studied extensively [13]. The best squeezing spectrum
for each field is
[1] For a recent review on squeezing see e.g. H. Bachor, “A
4a 1 GuidetoExperimentsin QuantumOptics”, Wiley-VCH
Sbest(ω)=1 , a= , (10) (2004).
j − (1+a)2+ω/κ2 2I [2] S.L. Braunstein, P. van Loock, Rev. of Mod. Phys. 77,
513 (2005).
yielding perfect squeezing at zero frequency at the turn- [3] P.Grangier,A.Levenson,J.-Ph.Poizat,Nature396,537
ing point where Y =1, I =0.5 and a=1. (1998).
ForY <1andI1 =I2 thefluctuationsofthetwofields [4] D. B.Horoshko, S.Ya. Kilin, Phys. Rev. A 61, 032304
6
are coupled. For I >I we get (2000).
1 2
[5] J-F.Roch,K.Vigneron,P.Grelu,A.Sinatra,J-P.Poizat,
1 η P. Grangier, Phys. Rev.Lett.78, 634 (1997).
δX = δX i − δY (11)
1 − 1− Y 1 [6] For a review on EIT see e.g. M. Fleischhauer, A.
1+η Imamoglu, J.P. Marangos, Rev. Mod. Phys. 77, 633
δY1 = δY1+i δX1 2iηδX2. (12) (2005);
− Y −
[7] K.M. Gheri, D.F. Walls, M.A. Marte, Phys. Rev. A 50,
Theequationsforfield2areobtainedfrom(11)and(12) 1871 (1994).
by changing the sign in front of η and of i. Simple ana- [8] V.A. Sautenkov, Y.V. Rostovstev, M.O. Scully, Phys.
Rev.A72,065801(2005),C.L.GarridoAlzar,L.S.Cruz,
lyticalexpressionscanbe obtainedforthe squeezingand
J.G. Aguirre G´omez, Europhys.Lett. 61, 485 (2003).
the conditional variance V of the fields at ω =0
sm [9] L.A.Lugiato Progess in Optics XXI, edited by E. Wolf
|
(North-Holland, Amsterdam, 1977), p. 71.
4
Sint = Sbest =1 ; Sphase = 3+ (13) [10] A.Sinatra,J-F.Roch,K.Vigneron,P.Grelu,J-P.Poizat,
j j j − η2 K. Wang, P. Grangier, Phys. Rev.A 57, 2980 (1998).
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Vs|m = 4−3η2 (14) PanhdysP..RGerva.nAgie4r2,,A2n9n9.5P(h1y99s.0)(;PaJr.Pish).1P9o,i2za6t5,(J1.9-F94.)R.och
showing that the fields have diverging phase noise and [12] K.M. Gheri, P. Grangier, J.-Ph. Poizat, D. Walls, Phys.
Rev. A 46, 4276 (1992).
become perfectly correlated at the turning point. We
[13] L. Lugiato, P. Galatola, L. Narducci, Opt. Comm. 76,
checked that the spectra in Fig. 5 are well reproduced
276 (1990).
by the analytical results.
[14] V. Josse, A. Dantan, A. Bramati, M. Pinard, E. Gia-
Inconclusion,inasymmetricallydetunedEITscheme, cobino, Phys. Rev.Lett.92, 123601 (2004);
andforequalinputintensitiesY ofthetwofieldswehave [15] J.K. Thompson, J. Simon, H. Loh, V. Vuleti´c, Science
shownthe existenceofa universalS-shapedsteady state 313, 74 (2006).