A single cold atom as efficient stationary source of EPR-entangled light

0.3 MB·English

by  David Vitali

A single cold atom as efficient stationary source of EPR-entangled light
David Vitali
CNISM and Dipartimento di Fisica, Universit`a di Camerino, 62032 Camerino, Italy
Giovanna Morigi
Grup d’Optica, Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain
Ju¨rgen Eschner
ICFO - Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
(Dated: February 1, 2008)
The Stokes and anti-Stokes components of the spectrum of resonance fluorescence of a single
trapped atom, which originate from the mechanical coupling between the scattered photons and
7
thequantized motion of the atomic center of mass, exhibit quantumcorrelations which are of two-
0
mode-squeezingtype. Westudyanddemonstratethebuild-upofsuchcorrelationsinaspecificsetup,
0
which is experimentally accessible, and where the atom acts as efficient and continuous source of
2
EPR-entangled,two-mode squeezed light.

n
a PACSnumbers: 42.50.Dv,32.80.Qk,32.80.Lg
J

6
2 I. INTRODUCTION In this manuscript we investigate the quantum corre-
lations between the Stokes and anti-Stokes sidebands of

2 the resonance fluorescence of a trapped atom, i.e. be-
The control of atom-photon interaction is object of
v
tweenthespectralcomponentswhichareduetothecou-
intensive research for its potentialities in quantum net-
1
pling of the electromagnetic field to the atom’s oscilla-
3 working. In fact, several experimental realizations have
tory motion [27, 28, 29, 30]. The spectrum is studied for
2 accessed novel regimes of engineering atom-photon in-
5 teractions and have opened promising perspectives for an atom tightly confined inside a resonatorand continu-
0 implementing controllednonlineardynamicswithsimple ously driven by a laser, in the setup sketched in Fig. 1.
6 This setup has been consideredin [25, 26] for the case of
quantum optical systems. Fundamental steps in this di-
0 pulsed excitation, where scattering could be considered
rection have been, amongst others, the generation of en-
/
h tangled light in atomic ensembles [1, 2], atomic memory coherent. In the present work, the atom is continuously
p for quantum states of light [3, 4, 5, 6, 7], and entan- driven and hence both coherent and incoherent scatter-
- ing processes determine the dynamics of the system. We
t glement of remote ensembles [8, 9, 10]. At the single
n find that in a suitable parameter regime the Stokes and
atom level, entanglement between a single atom and its
a anti-Stokes spectral components of the resonance fluo-
emitted photon [11] has been demonstrated in [12, 13],
u
rescencearetwo-modesqueezed,thatis, their amplitude
q while in cavity quantum electrodynamics generation of
and phase quadratures are quantum correlated. In fact,
: quantumlighthasbeenachieved,likelasingatthesingle
v the variance of the difference of the amplitude quadra-
atom level [14, 15], controlled single-photon generation
i
X [16, 17, 18, 19], as well as quantum state and entangle- tures of the two sideband modes, as well as the variance
of the sum of their phase quadratures, are squeezed be-
r ment engineering in the microwave regime [20].
a Quantumnetworkingwithsingletrappedatomsorions low the shot noise limit, hence reproducing the salient
properties of the entangled, simultaneous eigenstate of
shows severaladvantages, due to the high degree of con-
relative distance and total momentum of two particles,
trol one can achieve on these systems [19, 21, 22]. Con-
as considered in the original EPR paradox [31, 32]. In
trol can be gained on the internal as well as on the
our model, entanglement between the modes originates
external degrees of freedom, which can both be inter-
fromthemechanicalcouplingoftheelectromagneticfield
faced with light by exchange of angular and linear mo-
withthequantummotionoftheatom,anditisendorsed
mentum. In particular, by coupling the atomic external
by a specific setup, which achieves resonant emission of
degrees of freedom with photons via the mechanical ef-
the Stokes and anti-Stokes photons. In this regime, the
fect of light, atom-photon interfaces for continuous vari-
singleatomactsasanefficientcontinuoussourceofEPR-
ables can be implemented even at the level of a single
entangled, two-mode squeezed light.
atom [23, 24, 25, 26]. This concept has been specifically
applied in [25, 26], where the realization of a pulsed op- Conventionally,two-modesqueezedstatesemergefrom
tical parametricamplifier based ona single coldtrapped the nonlinear optical interaction of a laser with a crys-
atom inside a high-finesse optical cavity was proposed, tal, i.e. from parametric amplification or oscillation. As
and it was shown theoretically that this system allows such, the phenomenon is the result of many-atom dy-
for the controlled, quantum-coherent generation of en- namics, often described by a simple nonlinear polariza-
tangled light pulses by exploiting the mechanical effects tion model. In the single-atom case novel features ap-
of atom-photon interaction. pear which are due to the coherent microscopic dynam-
2
'
n
ics. Our study allows us to identify the dependence of
these features on the external parameters, thereby giv-
ingusinsightintohowmacroscopicpropertiesarisefrom
microscopic dynamics in this particular non-linear pro-
± '
cess. Moreover, we find peculiar spectral characteris- wL n
tics of the squeezing which are unique to this system,
and which we trace back to the interplay of the var-
ious time scales of the dynamics. In a more general
w
context, our study is connected to previous work on the L
quantum features of the spectrum of resonance fluores-
cence[27,28,33,34,35,36,37,38,39,40],andtorecent
experimentalandtheoreticalstudiesonquantumcorrela-
FIG. 1: Layout of the system. A single atom is confined by
tionsinthelightscatteredbyatoms[1,2,3,5,41,42,43],
an external potential inside an optical cavity and is driven
bysemiconductormicrocavities[44],andbymacroscopic by a laser. The cavity is resonant with the motion-induced
mirrors [45, 46]. Stokes and anti-Stokes components of the resonance fluores-
Thisarticleisorganizedasfollows. InSec.IIthebasic cence. Correlations between these spectral component are
coherent dynamics, giving rise to quantum correlations measured in the cavity output. The orientation of the con-
between the Stokes and anti-Stokes components of the sidered vibrational mode has non-zero projection onto the
spectrum of resonance fluorescence, are briefly reviewed, laser direction. A possible geometry to implement the sys-
tem would bean F =0 to F′ =1 atomic transition with the
andtheimportanttimescalesareintroduced. InSec.III
quantization axis B~ along the cavity axis, and B~, laser wave
the theoretical model is described in detail and the rele-
vector,andlaserpolarisationmutuallyorthogonal,andamo-
vantscatteringprocessesinthesystemareidentifiedand
tional mode parallel to the laser direction. More details can
discussed. In Sec. IV the spectrum of squeezing is eval-
be found in Refs. [25, 26] where pulsed coherent excitation
uated using Quantum Langevin Equations; for a quick
wasconsidered. Inthepresentpaperwedealwithcontinuous
overview of the main results without the full theoretical laser excitation.
elaboration,thereadermayfirstskipthis partandjump
to Sec. IVC where the squeezing characteristics are cal-
culated for a specific, experimentally achievable physical that χ2 > χ1 , with an angular frequency
| | | |
system. Finally, Sec. V presents the conclusions and an
outlook. Θ= χ2 2 χ1 2 . (2)
| | −| |
p
The time-evolution of the operators, in the Heisenberg
II. EPR-ENTANGLEMENT OF LIGHT AT THE representation, is given by [46]
CAVITY OUTPUT
χ 1
a (t) = 1b†(0)sinΘt+ χ 2 χ 2cosΘt a (0) ,
1 Θ Θ2 | 2| −| 1| 1
In this section we briefly review the coherent dynam- χ χ (cid:2) (cid:3)
ics, described previously in Refs. [25, 26], which lead to 1 2 [1 cosΘt]a†(0) (3)
− Θ2 − 2
two-mode squeezing between the Stokes and anti-Stokes χ χ χ
modes in the light scattered by a trapped, laser-driven a2(t) = Θ2b(0)sinΘt+ Θ122 [1−cosΘt]a†1(0)
atom. We thus first ignore incoherent processes and fo-
1
cus on the pulsed dynamics which can be obtained in −Θ2 |χ1|2−|χ2|2cosΘt a2(0) , (4)
a suitable parameter regime with a setup like the one (cid:2) (cid:3)
1
shown in Fig. 1. b(t) = b(0)cosΘt+ χ∗a (0)+χ a†(0) sinΘt .
Θh− 2 2 1 1 i
(5)
Thetrappedatomiscoupledtoanopticalcavityofwhich In general these solutions describe tripartite entangle-
two modes are resonant with the Stokes and anti-Stokes ment among cavity modes and center-of-mass oscilla-
sidebands,respectively. Forshorttimesthelaser-induced tor [46]. An interesting situation is found after half a
resonant interaction between the center-of-mass oscilla- period, for T = π/Θ. At this time (modulus 2π) the
π
tion, denoted by annihilation and creation operators b center-of-mass oscillator is uncorrelated with the cavity
and b†, and the two cavity modes, represented by oper- modes, which exhibit EPR-type entanglement [25, 26].
ators a and a† (j = 1,2), is described by the effective Clearly,thisdescriptionisapproximate,andvalidonly
j j
Hamiltonian in the interaction picture when incoherent processes can be neglected. In the
presentworkweconsiderthesituationinwhichtheatom
W =ih¯χ a†b†+ih¯χ a†b+H.c , (1) is continuously driven by the laser field, such that quan-
eff 1 1 2 2
tum noise and dissipative processes affect the dynam-
wherethescalarsχ indicatethestrengthofthecoupling. ics relevantly. We show that steady state entanglement,
j
This Hamiltoniangeneratesperiodic dynamics,provided i.e. quantum-correlated spectral fluctuations in the two-
3
mode cavity output field, is found also under these con- Here, H is the Hamiltonian for the relevant atomic de-
a
ditions. The details of this entanglement will depend on grees of freedom,
the comparison between the time scale set by the coher-
entdynamics,Θ−1,andthetimescalesofthedissipative Ha = ¯h∆e e +Hmec , (7)
− | ih |
processes, κ−1 for loss of photons from the cavity, and
γ−1 forspontaneousscatteringfromtheatom. Inpartic- where ∆=ωL ω0 is the detuning of the laser from the
−
dipole transition at the angular frequency ω , and
ular, we will show that the squeezing spectrum shows 0
distinct, qualitatively different features in the regimes
1
Θ < κ, Θ = κ, and Θ > κ. The reader is referred H =h¯ν b†b+ (8)
mec
(cid:18) 2(cid:19)
to Sec. IVC, where the spectra for different parameter
regimes are reported.
describes the harmonic motion of the atomic center of
mass at angular frequency ν, as determined by an ex-
ternalpotential, whereb,b† arethe annihilationandcre-
III. SCATTERING PROCESSES
ationoperators,respectively,ofaquantumofvibrational
energy h¯ν. In particular, the atomic position is givenby
The purpose of this section is to discuss the coherent x= ¯h/2Mν(b+b†). We denote by n the eigenstates
| i
and incoherent scattering processes determining the dy- of Hpmec at energy h¯ν(n + 1/2). The Hamiltonian for
namics of the system. We will present these processes thecavitymodes,whichcoupleappreciablytothedipole
using physical pictures derived from the scattering ma- transition, is
trixundermoderatesimplifications,inordertoillustrate
themorerigorousderivationspresentedinthesubsequent H = ¯hδ a†a , (9)
c − j j j
section. We first introduce the model, and then identify jX=1,2
thescatteringprocessesanddeterminethecorresponding
rates. whereδj =ωL ωj arethedetuningsofthelaserfromthe
−
frequencies ω of two optical modes, and a ,a† are the
j j j
respectiveannihilationandcreationoperatorsofaquan-
A. Model tumofenergyh¯ωj,i.e.aphotoninmodej. Wedenoteby
n ,n the eigenstates of H at energy ¯hδ n ¯hδ n ,
1 2 c 1 1 2 2
| i − −
andconsiderthesituationinwhichthemodefrequencies
We consider an atom of mass M inside an optical res-
fulfill the relation
onator and driven by a laser. The atomic motion is con-
fined by an external potential, which we assume suffi-
ω ω =2ν′ . (10)
ciently steep in the radial direction so that the motion 2− 1
in this plane can be considered frozen out. We denote
where
by x the axis of the remaining one-dimensional atomic
center-of-mass motion. Moreover, we assume that only ν′ =ν+δν (11)
the atomic dipole transition between ground state g
| i
andexcitedstate e couplesrelevantlytothefields,such andδν takesinto accountradiativeshifts, suchthat cav-
| i
thatwecanrestrictthe electronicdynamicstothese two ity modes 1 and 2 can be simultaneously resonant with
states. The atomic dipole is laser-driven, and it cou- the Stokes and the anti-Stokes transitions. This contri-
ples to two modes (j = 1,2) of the resonator, as well butionwillbediscussedinSec.IIIB4anddeterminedin
as to the external modes of the electromagnetic field. Sec. IVA.
The cavity modes couple also to the external modes of Finally,themodesoftheelectromagneticfieldexternal
the electromagnetic field through the imperfect mirrors to the cavity possess the free Hamiltonian
of the resonator. The total dynamics is governed by the
Hamiltonian H = ¯h δ r† r ¯h δ r† r ,
emf − kj kj kj − ks ks ks
Xkj Xks
H =H +W,
0
where r , r† are annihilation and creation operators,re-
λ λ
where H0 is the self-energy of the system of atom and spectively,ofaphotonatangularfrequencyωλ =ωL δλ,
fields, and W describes their mutual interaction, as well wavevector k and polarization e . Here, the subsc−ripts
λ λ
as the coupling between the cavity modes and the exter- λ=k and λ=k indicate the modes of the field which
s j
nal modes through the finite transmission at the cavity couple to the dipole and to the cavity modes (through
mirrors. We now introduce each term in detail, and dis- the mirrors), respectively. The interaction term
cuss the dynamics in the reference frame of the laser at
the angular frequency ωL. We decompose H0 according W =HaL+Hac+Wks +Wkj (12)
to
describes the couplings among atom and fields, decom-
H =H +H +H . (6) posed into four terms which correspond to the coupling
0 a c emf
4
between atom and laser (H ), atom and cavity modes limit all terms of W are weak perturbations to the dy-
aL
(H ), atom and modes of the external electromagnetic namics. Weassumethatthesystemisintheinitialstate
ac
field (W ), and cavity modes and external electromag-
ks
netic field (Wkj). We discuss these terms in the Lamb- |ψii=|g,n;01,02;0kj;0ksi, (15)
Dicke regime, when the atomic motion is well localized
with energy E = h¯νn, where the atom is in the ground
over the wavelengths of the fields, such that the Lamb- i
Dicke parameter η = ¯hk2/2Mν is small, η 1. At state g , the center-of-mass oscillator is in the number
| i
≪ state n ,andthecavitymodesandtheexternale.m.-field
lowest order in η, the cpoupling between laser and dipole | i
are in the vacuum state, 0 ,0 ;0 ;0 . The scattering
has the form [47] | 1 2 kj ksi
matrixelementsbetweentheinitialstateandallpossible
HaL = h¯Ωσ† 1 η2 cos2θL(2b†b+1) (13) final states |ψfi, with energy Ef, have the form
h(cid:18) − 2 (cid:19)
=δ 2πiδ(E E ) (16)
+ iηcosθ (b†+b)+O(η2) +H.c. , Sif if − f − i Tif
L
i
where δ is the Kronecker-delta, δ(E E ) is a delta-
if f i
with σ = g e the dipole lowering operator and σ† its −
function giving energy conservation between initial and
| ih |
adjoint, Ω the Rabi frequency, and θ the angle between
L final states, and is the transition matrix to be evalu-
if
thedirectionofpropagationofthelaserandthemotional T
ated in lowest order in perturbation theory,
axis xˆ. In what follows we denote the moduli of all rele-
vantwavevectorsbyk,astheirdifferencesarenegligible. 1
= ψ W ψ + ψ W W ψ
Thecouplingbetweenthedipoleandthe cavitymodesis Tif h f| | ii h f| E H | ii
i eff
represented by −
with
η2
Hac = h¯jX=1,2gjcosφjajσ†h(cid:18)1− 2 cos2θc(2b†b+1)(cid:19) Heff =−¯h(cid:16)∆+iγ2(cid:17)|eihe|+h¯νb†b−¯hjX=1,2(δj +iκj)a†jaj
ηcosθctanφj(b†+b) +H.c.+O(η2) , (14) (17)
− i Wenowconsiderallpossiblescatteringtransitionstores-
whereg isthecouplingstrengthofthedipoletomodej,
j onant states, i.e. to final states ψ at energy E =E .
f f i
and the cavity axis forms an angle θ with the axis xˆ of | i
c
themotion. Theangleφ takesintoaccounttheposition
j
of the trap center inside the standing wave of the cavity.
Finally, the terms
1. Scattering of laser photons into the external e.m.-field
η2
W = ¯hg σ†r 1 cos2θ (2b†b+1)
ks Xks ks ksh(cid:18) − 2 ks (cid:19) We consider the scattering of a laser photon
+ iηcosθ (b+b†)+O(η2) +H.c.) , into the external e.m.-field by spontaneous emission,
ks hence coupling of ψ to the final states ψ =
Wkj = Xkj ¯hgkj(a†jrkj +H.c.) i F|gi,gn.′;20(a1,).02;H0ekrje;,1ktshie.| cioiuTplhinisg wpriothcestsheiscavsiktyet|cmhkoesddieiins
neglected, as the cavity is far-detuned from the dipole,
describethecouplingofatomandcavitytothemodesof
and the rate of this process can be approximated by the
the external e.m.-field. Here, W is the coupling of the
ks scattering rate of the atom in free space,
dipole, atRabifrequenciesg ,withthe externalmodes,
ks
whose wave vectors form angles θ with the motional 1
axis. This coupling gives rise to theksfinite linewidth γ of Γsifp ≈ γ(cid:16)|ts0p|2δn′,n+|ts+p1|2(n+1) δn′,n+1 (18)
the excited state, γ =2πρ (ω )g (ω )2, with ρ (ω )
density of states of the e.mk.s-fie0ld|ckosupl0in|g to the aktsom0ic +|ts−p1|2n δn′,n−1(cid:17),
dipole at angular frequency ω . The term W describes
thecouplingofthecavitymod0eswiththeextekrjnalmodes where
at strengthg . This coupling gives rise to the linewidth
of the cavitykjmodes κj = π|gkj|2ρkj(ωj), with ρkj(ωj) ts0p = ∆+γΩiγ/2, (19)
density of states of the e.m.-field coupling to the cavity
cosθ cosθ
modes at angular frequency ωj. tsp =ηγΩ L + ks , (20)
+1 (cid:18)∆ ν+iγ/2 ∆+iγ/2(cid:19)
−
cosθ cosθ
B. Basic scattering processes ts−p1 =ηγΩ(cid:18)∆+ν+Liγ/2 + ∆+iγk/s2(cid:19) . (21)
Weconsiderthelimitinwhichtheatomisfar-detuned The process described by amplitude (19) does not af-
from cavity modes and laser, ∆ γ,δ ,g ,Ω. In this fect the dynamics of the cavity modes nor that of the
j j
| | ≫
5
(a) (b) (c)
FIG. 2: Basic scattering processes. (a): A laser photon is absorbed and emitted by the atom, without coupling to the cavity
mode. (b) A laser photon is scattered into the cavity mode and then rescattered by the atom into the external modes of the
electromagneticfield. (c)Alaserphotonisscatteredbytheatomintothecavitymode,andthenitistransmittedbythecavity
mirror into themodes of theexternal electromagnetic field.
center-of-mass motion. In contrast, the amplitudes (20) tcav =ηΩg∗cosφ icosθL cosθctanφj(2,5)
and (21) are coherent superpositions of scattering pro- j,+ j j(cid:20)∆ ν+iγ/2 − ∆+iγ/2 (cid:21)
−
cesses involving, respectively, the mechanical effect of icosθ cosθ tanφ
the laser and of the emitted photon on the atomic mo- tcj,a−v =ηΩgj∗cosφj(cid:20)∆+ν+Liγ/2 − ∆+c iγ/2j((cid:21)2.6)
tion [28, 30], thereby affecting the coherence of the mo-
tional state. Their rate is γ η2(cos2θ +α)γΩ2/∆2,
b L Like in Eqs. (20) and (21), we recognize on the RHS of
≈
where α describes the angular dispersion of the spon-
Eqs. (25) and (26) the coherent addition of two scatter-
taneously emitted photons, determined by the quantum
ing amplitudes, here representing the mechanical effects
numbers of the atomic transition [48].
ofthelaserandofthecavity,respectively[50,51]. These
processes are at the basis of the coherent coupling be-
tweentheatomicmotionandthecavitymodesdescribed
2. Scattering of laser photons into the cavity modes
by Hamiltonian (1), whereχ = itcav and χ = itcav.
1 − 1,+ 2 − 2,−
We are interested in the regime where energy can be
Next we discuss the processes in which a laser pho- stored in the cavity modes through this coupling, which
ton is scattered into one of the cavity modes, thereby requires χ , χ γ as a necessary condition. In this
1 2 b
coupling the initial state ψi to the states ψ1 = situation|,it|is|vis|ib≫lefromtheequationsthatinthelimit
t|gh,ense′;s1t1a,t0e2s;a0rkej;n0oktsistoarb|lψe,2bi|u=tir|egs,onn′;a0n1t,ly12c;o0ukpj;le0d|kstio.itAhes tκhje≪opνt,imbyumchoeonshinagncδe1m=enνtaonfdtδh2e=sc−atνteorninegcaonf aachlaiesveer
continuumofstates|g,n′;01,02;1kj;0ksibycavitydecay, photon into mode 1 accompanied by the excitation of
the correct final states of these scattering processes de- the motion by one vibrationalquantum, and of the scat-
scribe the processes sketched in Fig. 2(c) and have the tering ofa laserphotoninto mode 2 accompaniedby the
form de-excitationof the motion by one vibrationalquantum.
Q Note that these scattering terms, tcav, have an incoher-
j j,±
ψ = Z 1+ W ψ , (22)
| kji q kj(cid:18) Ekj −H kj(cid:19)| ji einntgecnoemraplocnoehnetrwenhticdhynscaamleicsswciatnhoγn/l|y∆b±e aνc|h.ieTvheedrewfohreen,
where Qj projects onto the subspace orthogonalto ψj , γ ∆, on a time scale such that incoherent terms are
| i ≪| |
and Zkj ensures the normalizationof the state. Further- negligible. Moreover, the condition γ ≪ ν is also re-
more, Zkj gives the occupation probability of state |ψji, quired in order to create quantum correlations between
since Z = ψ ψ 2. the two cavity modes, since the difference between the
kj |h j| kji|
The coupling rate between state |ψii and states |ψkji two coupling strengths χ1 and χ2 determines the typi-
takes the form cal time scale on which entanglement is established, see
Sec. II and [26].
2κ
Γcifajv ≈ δ2+jκ2 |tc0av|2 δn′,n (23)
j j
2κ
+(δ ν)j2+κ2 |tcj,a+v|2(n+1) δn′,n+1 3. Scattering of cavity photons into the external e.m.-field
j − j
+(δ +2νκ)j2+κ2 |tcj,a−v|2n δn′,n−1, intAostshuemicnagvitthyatmpohdoetso,ntshheyavceabneebnecroeh-aerbesnotrlbyesdcabtyterthede
j j
atom and emitted spontaneously into the external e.m.-
where
field, as sketched in Fig. 2(b). In order to focus on the
1 evaluationofthecorrespondingelementofthescattering
tcav =Ωg∗cosφ , (24)
0 j j∆+iγ/2 matrix, we consider the regime of very small cavity loss
6
rate, i.e. we assume stable cavity modes and ignore, for 1 1 1
Re +
the clarity of the picture, cavity decay. Be the initial × (cid:26)∆+ν+iγ/2 ∆ ν+iγ/2 − ∆+iγ/2(cid:27)
−
state = η2cos2θ Ω2 b†b
L
ψ = g,n;m ,m ;0 ;0 (27) 2∆(∆2 ν2+γ2/4) ∆
| i,mi | 1 2 kj ksi − .
× (cid:18)(∆2 ν2+γ2/4)2+γ2ν2 − ∆2+γ2/4(cid:19)
at energy E = h¯νn ¯hm δ ¯hm δ , with the atom −
i,m 1 1 2 2
− −
in g , the center-of-mass oscillator in the number state Finally, off-resonant coupling of the cavity mode with
| i
n , the cavity modes in the Fock states m and m , the dipole transition gives rise to an a.c.-Stark shift of
1 2
| i | i | i
andtheexternale.m.-fieldinthevacuumstate, 0 ;0 . the cavity mode levels, which reads at leading order
| kj ksi
This state is coupled to the states
g 2cos2φ (∆ δ )
δω | j| j − j a†a +O(η2). (32)
|ψf,m′1i=|g,n;m1−1,m2;0kj;1ksi (28) j ≈ (∆−δj)2+γ2/4 j j
|ψf,m′2i=|g,n;m1,m2−1;0kj;1ksi (29) Theseshiftsareingeneralnotsmallandshouldbetaken
into account, when aiming at the resonant enhancement
by absorption of a cavity photon and spontaneous emis-
of certain processes over others. It should be remarked
sion. We evaluate the corresponding rate under the as-
that the correctionto δω in Eq. (32) which is at second
sumption, that tanφ = 0, i.e., there are no mechanical j
j order in η arises from the mechanical effects of the in-
effects of the resonator on the atom at first order in η,
teraction between resonator and center-of-mass motion.
and find an effective loss rate of the cavity modes
This term is non-linear,as it is a shift which depends on
Γicfajv−sp = γ|gj|2(cid:12)(cid:12)∆−√δjm+jiγ/2(cid:12)(cid:12)2 . (30) rtchiosenetnrtoiubmuanbtieoarndodtfoitviδoibωnrjaa.ltOicoonnnattlrhieebxuoctitithoaentriothona,tnbhdue,ttahi.scis.-aStentaremrgkliggsiihvbielfest
(cid:12) (cid:12)
It should be noted that(cid:12)these processe(cid:12)s arise from of the center-of-mass motion, which is of the same order
atomicscatteringofalaserphotonintothecavitymodes, as δνb and depends on the number of photons. Its effect
whichisthenrescatteredbyatomicemissionintotheex- is detrimental, as the resulting spectrum of the center-
ternal modes of the e.m.-field. Hence, these processes of-mass excitations deviates from the one of a harmonic
can interfere with atomic scattering of a laser photon, oscillator. In the system we consider we will neglect
in the limit discussed in Sec. IIIB1, in which the cou- this contribution,focussing ontothe regimein whichthe
plingtothecavityplaysnorole. Inthesecalculationswe mechanical effects of the cavity mode can be neglected.
have not considered the coherent addition of these two This correspondsto situations, wherethe motion, forin-
noise effects, but we will consider phase relations and stance, is almost orthogonal to the cavity wave vector,
possible interference in these noise sources when study- cosθc 1.
| |≪
ing the dynamics with the quantum Langevin equations
in Sec. IVA.
IV. SPECTRUM OF LIGHT AT THE CAVITY
OUTPUT
4. a.c.-Stark shift of the ground state energy
In this section we evaluate the spectrum of the light
Since the efficiency of production of two-mode transmitted by the cavity mirror. The spectrum is best
squeezed light is based on the resonant enhancement of evaluatedusing the quantum Langevinequations for the
two-photonprocesses,itisimportanttoconsidersystem- operators aj, a†j and b. The equations we obtain are
aticallyradiativecorrectionstotheresonancefrequencies rather involved, however the physical meaning of each
inthe implementationofthe dynamicsinSec.II. There- term can be identified by comparison with the rates of
fore, we now evaluate corrections to the energy of state thescatteringprocessesdiscussedintheprevioussection.
ψ , Eq. (27), due to far-off resonance coupling in the
i,m
| i
limit of very small cavity decay rates. When considering
thea.c.-Starkshiftofstate ψ ,wefindthreecontribu- A. Quantum Langevin Equations
i,m
| i
tions, each associated to a different kind of coupling: (i)
the a.c.-Starkshift due to the off-resonantlasercoupling We shall study the dynamics using the quantum
withthe excitedstateatzeroorderinthe mechanicalef- Langevin equations (QLE) of the system. For conve-
fects, δω Ω2/∆ for ∆ γ. It leads to a shift δω of nience,wewritetheinteractionHamiltonianofthe atom
0 0
∼ | |≫
the dipole resonance frequency. The mechanical effects with the laser and the cavity fields as
of the laser on the atoms give rise to (ii) a contribution
H =H +H =h¯ σ†B+σB† , (33)
which is linear in the number of vibrational excitation, int aL ac
(cid:0) (cid:1)
andcanhencebeconsideredarenormalizationofthetrap where
frequency. This a.c.-Stark shift reads
η2
B =Ω 1 cos2θ (2b†b+1) +iηΩcosθ (b†+b)
δνb η2cos2θLΩ2b†b (31) (cid:18) − 2 L (cid:19) L
≈
7
+ gjcosφjaj 1 η2 cos2θc(2b†b+1) hfin(t)fin(t′)†i=δ(t−t′), (41)
jX=1,2 (cid:18) − 2 (cid:19) hbin(t)bin(t′)†i= N¯ +1 δ(t−t′), (42)
ηcosθ g sinφ a (b†+b), (34) bin(t)†bin(t′) =N(cid:0)¯δ(t (cid:1)t′), (43)
− c j j j h i −
The QLE read where N¯ is mean thermal vibrational number of the ef-
fective thermal reservoir coupling to the atom center-of-
a˙ (t) = iδ a (t)+iσ(t) B(t)†,a (t)
1 1 1 1 mass motion [52].
−κ1a1(t)+√2κ(cid:2)1ai1n(t), (cid:3) (35) Weassumethatthelaserisred-detunedandfar-offres-
a˙ (t) = iδ a (t)+iσ(t) B(t)†,a (t) onancefromtheatomictransition,i.e.,∆isnegativeand
2 2 1 2
∆ is much larger than all the other parameters. This
κ a (t)+√2κ(cid:2)ain(t), (cid:3) (36) | |
− 2 2 2 2 allows us to eliminate adiabatically the atomic internal
b˙(t) = iνb(t)+iσ(t) B(t)†,b(t) +iσ(t)†[B(t),b(t)] degrees of freedom, and to assume that the atom always
−
κbb(t)+√2κb(cid:2)bin(t), (cid:3) (37) remains in the ground state |gi, that is, σz(t) ≈ −1.
− Therefore we neglect the time evolution of σ , Eq. (39),
γ z
σ˙(t) = i∆ σ(t)+σz(t) iB(t)+√γfin(t) ,(38) while Eq. (38) becomes
h − 2i (cid:0) (cid:1)
σ˙z(t) = 2iσ(t)B(t)† 2iσ†(t)B(t) γ[σz(t)+1] γ
− − σ˙(t)= i∆ σ(t) iB(t) √γfin(t), (44)
−2σ†(t)√γfin(t)−2σ(t)√γfin(t)†, (39) −(cid:16)2 − (cid:17) − −
where σz =σ†σ σσ†, and we have introduced the vac- whose formal solution is
uum input noise−s ain(t) (j = 1,2) of the cavity modes
j
withcorrespondingdecayrateκj,the spontaneousemis- σ(t)=e−(γ2−i∆)tσ(0) (45)
sion noise fin(t) at rate γ, and we also added a phe-
t
nomenologicalinputnoisebin(t)actingontheatom’smo- dse−(γ2−i∆)s iB(t s)+√γfin(t s) .
tion, describing the heating at rate κb due to the fluctu- −Z0 (cid:2) − − (cid:3)
ationsofthe trappotential. Thesefour noisesourcesare
mutually uncorrelated and have zero mean, while their We now insert solution (45) into the other QLE and ne-
second-order correlations have the form glect the transient term because we are interested in the
dynamicsattimeswhicharemuchlargerthan1/∆. We
ain(t)ain(t′)† = ain(t)ain(t′)† =δ(t t′), (40) obtain | |
h 1 1 i h 2 2 i −
t
a˙1(t) = iδ1a1(t)+Z dse−(γ2−i∆)s B(t−s)−i√γfin(t−s) B(t)†,a1(t) −κ1a1(t)+√2κ1ai1n(t), (46)
0 (cid:2) (cid:3)(cid:2) (cid:3)
t
a˙2(t) = iδ2a1(t)+Z dse−(γ2−i∆)s B(t−s)−i√γfin(t−s) B(t)†,a2(t) −κ2a2(t)+√2κ2ai2n(t), (47)
0 (cid:2) (cid:3)(cid:2) (cid:3)
t
b˙(t) = iνb(t)+ dse−(γ2−i∆)s B(t s) i√γfin(t s) B(t)†,b(t)
− Z − − −
0 (cid:2) (cid:3)(cid:2) (cid:3)
t
dse−(γ2+i∆)s B(t s)†+i√γfin(t s)† [B(t),b(t)] κbb(t)+√2κbbin(t), (48)
−Z − − −
0 (cid:2) (cid:3)
wherewehavenottakencareofoperatorordering,since, account the a.c.-Stark shifts due to the mechanical cou-
asweshallsee,withinthe validitylimitofourtreatment plingwithlaserandcavitymodes,seeSec.IIIB4,sothat
these integral terms will generate only linear contribu- thetwocavitymodesareresonantwiththemotionalside-
tions. bands of the laser light. Together with this choice of the
At this point, we choose the laser angular frequency laser frequency, we assume that the motional sidebands
ω so that are well resolved, that is, ν g ,Ω,κ .
L j j
≫| |
δ =ν′ ; δ = ν′ In order to identify the resonant process, we move to
1 2
− a frame rotating at the effective vibrational angular fre-
namely, the laser frequency is tuned symmetrically be- quency ν′ ν, (which has to be determined by solving
≃
tweenthemodefrequencies,whicharespacedbyaquan- the QLE) and we will neglect in the QLE all the terms
tity 2ν′. The angular frequency ν′ ν, and takes into oscillating at ν′ or larger. Denoting the slowly vary-
≃
8
ing quantities by a˜†(t) eiν′ta†(t), a˜ (t) eiν′ta (t), tors we obtain
˜b(t) eiν′tb(t), afte1r exp≡licitly ev1aluati2ng th≡e comm2uta-
≡
t
a˜˙†1(t)=i(ν′−δ1)a˜†1(t)+Z0 dse−(γ2+i∆)shB(t−s)†eiν′t+i√γfin(t−s)†eiν′ti (49)
g cosφ 1 η2 cos2θ (2˜b†˜b+1) +ηg sinφ cosθ ˜b(t)e−iν′t+˜b†(t)eiν′t κ a˜†(t)+√2κ a˜in(t)†,
×(cid:20)− 1 1(cid:18) − 2 c (cid:19) 1 1 c(cid:16) (cid:17)(cid:21)− 1 1 1 1
t
a˜˙2(t)=i(ν′+δ2)a˜2(t)+ dse−(γ2−i∆)s B(t s)eiν′t i√γfin(t s)eiν′t (50)
Z0 h − − − i
g∗cosφ 1 η2 cos2θ (2˜b†˜b+1) +ηg∗sinφ cosθ ˜b(t)e−iν′t+˜b†(t)eiν′t κ a˜ (t)+√2κ a˜in(t),
×(cid:20)− 2 2(cid:18) − 2 c (cid:19) 2 2 c(cid:16) (cid:17)(cid:21)− 2 2 2 2
t
˜b˙(t)=i(ν′ ν)˜b(t)+ dse−(γ2−i∆)s B(t s)eiν′t i√γfin(t s)eiν′t (51)
− Z0 h − − − i
iηΩ∗cosθ +ηg∗sinφ cosθ a˜†(t)e−iν′t+ηg∗sinφ cosθ a˜†(t)eiν′t
×h L 1 1 c 1 2 2 c 2 i
t
dse−(γ2+i∆)s B(t s)†eiν′t+i√γfin(t s)†eiν′t (52)
−Z0 h − − i
iηΩcosθ +ηg sinφ cosθ a˜ (t)eiν′t+ηg sinφ cosθ a˜ (t)e−iν′t κ ˜b(t)+√2κ ˜bin(t), (53)
L 1 1 c 1 2 2 c 2 b b
×h− i−
where we have introduced the noise operators a˜in(t) makingtheMarkovianapproximationexp (γ/2 i∆+
e−iν′tain(t), a˜in(t) eiν′tain(t), and˜bin(t) eiν′1tbin(t≡), imν′)s δ(s)/(γ/2 i∆ + imν′), for{m− = ±1,0,1.
which a1re still2delta≡-correla2ted. ≡ After l}on≈g, but straigh±tforward calculations we−get the
We insert in these equations the explicit expression final, effective QLE at leading order in the Lamb-Dicke
for B(t s), thereby neglecting the terms oscillating at parameter, which read
ν′ or fa−ster. We finally perform the time integrals by
a˜˙†(t) = i(ν′ δ )a˜†(t)+χ∗˜b(t) (κ +κ iδ )a˜†(t)+√2κ a˜in(t)†+√2κ¯ a˜in(t)†+F , (54)
1 − 1 1 1 − 1 1L− 1L 1 1 1 1L 1L 1
a˜˙ (t) = i(ν′+δ )a˜ (t)+χ ˜b(t) (κ +κ +iδ )a˜ (t)+√2κ a˜in(t)+√2κ¯ a˜in(t)+F , (55)
2 2 2 2 − 2 2L 2L 2 2 2 2L 2L 2
˜b˙(t) = i(ν′ ν)˜b(t)+χ¯ a˜†(t) χ¯∗a˜ (t) (κ +κ κ +iδ )˜b(t) (56)
− 1 1 − 2 2 − b 2b− 1b b
+√2κ ˜bin(t)+√2κ¯ a˜in(t) √2κ¯ a˜in(t)†+F .
b 2b 2L − 1b 1L b
Letusnowdiscusseachtermappearingintheequations. (59)
The coupling coefficients are given by cosθ itanφ cosθ
χ¯ =ηΩg∗cosφ L + 2 c ,
2 2 2(cid:18)∆+ν′ iγ/2 ∆+iγ/2 (cid:19)
−
χ =ηΩg∗cosφ cosθL + itanφ1cosθc , (60)
1 1 1(cid:18)∆ ν′+iγ/2 ∆+iγ/2 (cid:19)
−
(57) and correspond to the Raman processes, in which laser
photonsarescatteredintothecavitymodewithachange
cosθ itanφ cosθ
χ2 =ηΩg2∗cosφ2(cid:18)∆+ν′+Liγ/2 + ∆+2iγ/2 c(cid:19) , in the center-of-mass excitation, see Sec. IIIB2.
New fluctuation-dissipation sources appear in the
(58)
equations. We first discuss noise terms appearing in
χ¯ =ηΩg∗cosφ cosθL + itanφ1cosθc , Eqs. (54) and (55). In addition to cavity decay with
1 1 1(cid:18)∆ ν′ iγ/2 ∆+iγ/2 (cid:19) rates κj we find processes described by the decay terms
− −
9
with rate κ and κ , and the corresponding Langevin atomicmotionatrateκ κ ,to a finaleffective mean
1L 2L 2b 1b
noises a˜in(t) and a˜in(t), where vibrational number n =−κ /(κ κ ) ∆/4ν′, as
1L 2L th 1b 2b− 1b ≃ | |
in standard cooling [48]. However, the noise associated
γ g1 2cos2φ1 with these incoherent phonon absorptions and emissions
κ1L = 2γ2/|4+| (∆ ν′)2, (61) is correlated with the noise terms a˜in(t) and a˜in(t) de-
− 1L 2L
γ g 2cos2φ scribing scattering of cavity photons, because all these
2 2
κ2L = 2γ2/|4+| (∆+ν′)2, (62) processes ultimately originate from spontaneous emis-
sion. This is why the noise terms in the Langevin equa-
and tionfortheatomicmotionaredirectlyexpressedinterms
of a˜in(t) and a˜in(t), making therefore this correlation
1L 2L
a˜in(t)=fin(t)e−iν′t, (63) evident.
1L
a˜in(t)=fin(t)eiν′t. (64) TheoperatorsFj inEqs.(49)-(53)representnon-linear
2L terms, which describe the noise associated with the in-
coherent part of the scattering processes discussed in
Thesenoisesdescribeinput-outputprocessesbetweenthe
Sec. IIIB2. These terms can be neglected with respect
cavitymodesandexternalmodes,mediatedbytheatom.
to the coherent processes, provided that γ ∆ and
They possess the same correlationfunctions of the spon-
≪ | |
taneous emission noise fin(t), and at the timescales of γ ν. In particular, the second inequality ensures that
≪
interest, ν′t 1, they are uncorrelated from each other, rates χ1 and χ2 differ appreciably, such that entangle-
≫ ment between the cavity modes can be established in a
thanks to the oscillating factors. Note that
finite time [26]. We will focus on this regime, γ ν, in
≪
γ g1cosφ1 which we can thus neglect Fj in the effective QLE when
κ¯1L =−ir2γ/2+i(∆ ν′), (65) evaluating the spectrum of squeezing.
− Finally, the frequency shifts of the two cavity modes
γ g∗cosφ
κ¯ =i 2 2 , (66) and of the vibrational motion read
2L r2γ/2 i(∆+ν′)
− (∆ ν′)g 2cos2φ
1 1
δ = − | | (71)
with κjL = κ¯jL 2. They originate from the scatter- 1L γ2/4+(∆ ν′)2
ing processes|in w|hich cavity photons are lost because −
(∆+ν′)g 2cos2φ
2 2
they are absorbed and then spontaneously emitted by δ2L = γ2/4+|(∆|+ν′)2 (72)
the atom, as has been discussed in Sec. IIIB3.
The noise and dissipation terms in Eq. (56), in addi- 2∆η2 Ω2cos2θL γ2/4+∆2 ν′2
δ = | | − (73)
tion to the noise terms of the trap, are described by the b (γ2/4+∆2 (cid:0)ν′2)2+ν′2γ2 (cid:1)
decaytermswithrateκ andκ ,andthecorresponding −
1b 2b ∆
Langevin noise operators a˜in(t) and a˜in(t). These pro- η2 Ω2cos2θ
1L 2L − | | L∆2+γ2/4
cesses originate from incoherent emission or absorption
ofavibrationalquantumaccompaniedbyabsorptionand
and from their form one can recognize the a.c.-Stark
subsequent spontaneous emission of a laser photon. The
shifts reported in Sec. IIIB4, with δω = δ a†a ,
emission of vibrational quanta takes place at rate j jL j j
Eq. (32), and δν = δ b†b, Eq. (31), where now ν ν′.
b b
→
γ η2 Ω2cos2θ Notethatwehaveomittedanon-linearshiftatsecondor-
L
κ2b = 2γ2/4|+|(∆+ν′)2, (67) derintheLamb-Dickeparameter,whichaffectsbothcav-
itymodesandmotion. AsdiscussedinSec.IIIB4,thisis
while the rate of incoherent absorption of vibrational asmallcorrectiontoδ ,asitscaleswithη2,whileitmay
jL
quanta is given by havea relevanteffect onthe center-of-massdynamics. It
can be neglected in the limit Ωcos2θ g cos2θ . Un-
L j c
γ η2 Ω2cos2θL der this assumption, which we will con≫sider in the rest
κ = | | . (68)
1b 2γ2/4+(∆ ν′)2 of this manuscript,the spectrum ofthe center-of-massis
−
the spectrum of a harmonic oscillator, characterized by
Inparticular,when∆<0thenκ2b >κ1bandthemotion equidistant energy levels.
is cooled. Moreover, Asthe dynamicsweseekreliesonresonantinteraction
betweenthecavitymodesandthevibrationalmotion,the
γ ηΩcosθ
κ¯ = L , (69) twocavitymodesshouldbeexactlyatresonancewiththe
1b r2γ/2+i(∆ ν′) sidebands of the driving laser. Equation (73) provides
−
γ ηΩ∗cosθ an implicit equation for the actual vibrational angular
L
κ¯2b =r2γ/2 i(∆+ν′), (70) frequency ν′. In the parameter regime η Ω ∆,ν we
| | ≪ | |
− find with good approximation
with κ = κ¯ 2. If we consider the dynamics described
jb jb
by these te|rms| only, these incoherent phonon absorp- ν′ ν+ 2∆η2|Ω|2cos2θL γ2/4+∆2−ν2
tionandemissionprocessesleadtothermalizationofthe ≈ (γ2/4+∆2 (cid:0)ν2)2+ν2γ2 (cid:1)
−
10
∆
η2 Ω2cos2θ . (74) Inthe parameterregimeγ ν, using conditions (75)-
− | | L∆2+γ2/4 (76), we arrivethereforetoth≪e finalQLE,describingthe
coherent interaction between the two cavity modes and
TakingalsointoaccountthefrequencyshiftsofEqs.(71)- the vibrational motion, competing with losses and noise
(72), the resonance conditions are finally processesduetospontaneousemission,cavitydecay,and
vibrational heating,
δ =δ +ν′ (75)
1 1L
δ =δ ν′. (76)
2 2L
−
a˜˙†(t)=χ∗˜b(t) (κ +κ )a˜†(t)+√2κ a˜in(t)†+√2κ¯ a˜in(t)† , (77)
1 1 − 1 1L 1 1 1 1L 1L
a˜˙ (t)=χ ˜b(t) (κ +κ )a˜ (t)+√2κ a˜in(t)+√2κ¯ a˜in(t) , (78)
2 2 − 2 2L 2 2 2 2L 2L
˜b˙(t)=χ¯ a˜†(t) χ¯∗a˜ (t) (κ +κ κ )˜b(t)+√2κ ˜bin(t)+√2κ¯ a˜in(t) √2κ¯ a˜in(t)† . (79)
1 1 − 2 2 − b 2b− 1b b 2b 2L − 1b 1L
B. Evaluation of the spectrum of squeezing chosen normalization for the output cavity modes at ω,
the sum criterion reads
We now use Eqs. (77)-(79) in order to determine the
S (ω)+S (ω)<2, (85)
stationaryspectrumofsqueezingofthelightatthecavity + −
output. We consider
while the product criterion gives
Iout(t) = aout(t)+aout(t)† aout(t) aout(t)†, (80)
− 1 1 − 2 − 2 S+(ω)S−(ω)<1, (86)
Iout(t) = i aout(t) aout(t)†+aout(t) aout(t)†(81,)
+ − 1 − 1 2 − 2 so that in our case both criteria imply that the two out-
(cid:2) (cid:3)
corresponding respectively to the difference between the put modes are EPR-like entangled as soon as S(ω) < 1.
amplitudequadratures,andthesumofthephasequadra- The squeezing spectrum S(ω) can be obtained from the
tures of the two sideband modes. These are the quadra- Fourier transform of the Langevin equations after long
turesexhibitingtwo-modesqueezinginthecaseofpulsed but straightforward algebra, yielding a cumbersome ex-
excitation in this setup, see [25, 26]. The output cav- pressionwhichwillnotbereportedhere. Thisexpression
ity fields aojut(t) in Eqs. (80)-(81) are given by the usual becomes considerably simpler in the limit |Ω|,|gj|,γ ≪
input-output relation ∆ and η 1. In this limit the additional loss pro-
| | ≪
cesses due to spontaneous emission, associated with the
aojut(t)= 2κjaj(t)−aijn(t), j =1,2. (82) ratesκjL andκjb (j =1,2),aretypicallynegligible,that
p is, κjL,κjb κ. Moreover, we consider the case of ion
≪
The spectrum ofsqueezing canbe calculatedby evaluat- traps, where heating of the atomic motion is negligible
ing the Fourier transforms with respect to all radiative noise sources [56]. Finally,
asthe twocavitymodes areverycloseinfrequency,they
Iˆout(ω)= dteiωtIout(t), (83) will have very similar properties, in particular we can
± ±
Z take κ = κ = κ. In this parameter regime the main
1 2
aspects of the squeezing spectrum can be grasped from
and using the fact that at the stationary state it is
its analytical expression. One finds
hIˆ±out(ω)Iˆ±out(ω′)+Iˆ±out(ω′)Iˆ±out(ω)i=8πS±(ω)δ(ω+ω′), κ2 Θ4 Σ4
(84) S(ω)=1 − , (87)
where we havenormalizedthe spectrumso that the shot − (κ2+ω2) ((cid:0)ω2 Θ2)2(cid:1)+ω2κ2
h − i
noiselevelcorrespondstoS (ω)=1. Two-modesqueez-
±
ing is found when one spectrum of squeezing takes val- where Θ= χ 2 χ 2 as given in Eq. (2), and
2 1
| | −| |
ues below the shot noise limit at some ω. From the p
Fourier transform of Eqs. (77)-(79) one can see that
2 2
Σ= χ + χ 2χ χ , (88)
S+(ω) = S−(ω) ≡ S(ω), which implies that in the r(cid:12)| 2| | 1| − 1 2(cid:12)
present case two-mode squeezing is equivalent to EPR- (cid:12) (cid:12)
(cid:12) (cid:12)
like entanglementbetween the two output cavity modes. and we have used that χ = χ¯ when γ ∆ (see
j j
≪ | |
Thisiseasilyverifiedbyapplyingasufficientcriterionfor Eqs. (57)-(60)). Note that due to the transformations
entanglement,suchas the “sum” criterionofDuan et al. which we have applied, the results which appear around
[53], or the product criterion of Ref. [54, 55]. With the ω = 0 in S(ω) describe quantum correlations of noise