The Linewidth of Ramsey Laser with Bad Cavity

0.06 MB·English

by  Yang Li

.
The Linewidth ofRamseyLaserwithBad Cavity
Yang Li, Wei Zhuang, Jinbiao Chen, and Hong Guo
∗ †
CREAM Group, State Key Laboratory of Advanced Optical Communication
Systems and Networks (Peking University) and Institute of Quantum Electronics,
School of Electronics Engineering and Computer Science,
andCenterforComputationalScienceandEngineering(CCSE),PekingUniversity,Beijing100871, P.R.China
(Dated:January15,2010)
WeinvestigateanewlaserschemebyusingRamseyseparated-fieldtechniquewithbadcavity. Bystudying
thelinewidthofthestimulated-emissionspectrumofthiskindoflaserinsidethecavity,wefinditslinewidth
0 is more than two orders of magnitude narrower than atomic natural linewidth, and it is far superior to that
1 ofconventional opticalRamseymethodandanyotheravailablesubnatural linewidthspectroscopy atpresent.
0 Sinceanycavityrelatednoiseisreducedtocavity-pullingeffectinbadcavitylaser,thisRamseylaserprovides
2 thepossibilityofprecisionsubnaturallinewidthspectroscopy,whichiscriticalforthenextgenerationofoptical
clockandatominterferometers.
n
a
PACSnumbers:42.55.Ah,42.50.Ar,42.60.Da,32.30.-r
J

5
1 Introduction: Since the invention of the separated-field cus on the stimulated emission spectrum via multiple coher-

technique [1], it has played an important role in the field of ent interactionsinside the cavity. We find this Ramsey laser
]
h precision spectroscopy due to its linewidth narrowing effect canprovideastimulated-emissionspectrumwithalinewidth
p via multiple coherent interaction. Atomic clocks based on muchnarrowerthanthatofanyconventionalopticalRamsey
-
thistechniquehavegreatlyextendedourabilityforfrequency seperated-field spectroscopy, which is commonly applied in
t
n measurement,further,almostalltheatominterferometersare opticalatomicclock. Ourresultsalsoshowthatasubnatural
a
basedonthistechnique[2]. linewidthspectroscopy,superiortoanyotheravailablesubnat-
u
q Though, the natural linewidth of quantum transition was uralspectroscopytechniqueatpresent[3–10],canbereached
[ regarded as the ultimate limit to high-resolution laser spec- bythiskindoflaser,ifasuitableatomiclevelstructureischo-

1 troscopy [4], several methods of subnatural linewidth spec- sen. Thus, this method can provide an effective subnatural
v troscopyhavebeenproposedtogainsubnaturallinewidth[3– spectroscopy, and the possibilities for the new optical clock
0 10]. However, in all these efforts, including optical Ramsey scheme[15]andatominterferometers[2].
7 spectroscopy, subnatural line is realized at the expense of a Theoreticalframework:Weconsiderthecaseofatwo-level
6
quickreductionin signal-to-noise(SNR) ratiodueto the ex- atomic beam interacting with a single-mode Ramsey cavity
2
. ponentialdecayingofsignal,thusalltheseschemescanonly ofseparated-oscillating-fieldresonatorswiththecavitymode
1 getthelinewidthseveraltimesnarrowerthantheatomicnat- linewidthismuchwiderthantheatomicgainlinewidth.Thus
0
ural linewidth. In the past three decades, this situation does we call it bad-cavity Ramsey laser. All atoms are pumped
0
1 not change in the field of the precision laser spectroscopy. onto the upper lasing state a before entering the first cavity
: On the other hand, the thermalnoise of the cavity mirrorsis ofseperatedfield,andthelowerlasingstateisb. Weassume
v
the main obstacle for further linewidth reduction of a laser alltheatomshavethesamevelocitiesυ,thatmeanswhatwe
i
X [11,12],anditisachallengetosubstantiallyreducethisnoise consider here is a homogeneous laser system. And for the
r further[13]. Recently, a new scheme, called active optical sakeofsimplicity,weconsiderthetwo-standingwaveslinear
a
clock[14–18],wasproposedtosubstantiallyreducethelaser optical Ramsey configuration with a grid as spatial selector
linewidth. With lattice trappedatoms, it is possible to reach [20, 21]. Our treatment can be extended to other configura-
mHzlinewidthlaserbasedonthemechanismofactiveoptical tions as in [22–24]. The length of each oscillating part is l,
clock[14,15,19]. Theprincipalmechanismofactiveoptical andthelengthofthefreedriftregionisL. Thecorresponding
clockistodirectlyextractlightemittedfromtheultranarrow Hamiltonianis
atomic transition with a cavity mode linewidth much wider
thanthatoflasing.Thisbadcavityensuresthatanyfrequency H = ~ωaˆ†aˆ +~ [ωaj(t)σaj +ωbj(t)σbj]
shiftduetocavitynoisereducestocavity-pullingeffect[15– Xj
17],thenthethermalnoiseisnotthemajorobstacleagainfor + ~g Γj(t)(aˆ†σˆje−i~k·~rj +σˆ+jaˆei~k·~rj), (1)
reducingthelinewidth.Thismeansthebadcavitycanplayan Xj −
indispensableroleinnewsubnaturallinewidthspectroscopy.
In this Letter, we propose a new scheme called Ramsey whereaˆ,aˆ aretheannihilationandcreationoperatorsofthe
†
laserwithbadcavity. Distinctfromanypreviousapplications field mode inside the cavity, with the frequency ω, σj =
a
of conventional Ramsey separated oscillating fields method (a a)jandσj =(b b)jaretheprojectionoperatorsforthe
| ih | b | ih |
[1], which focuses on the absorption spectrum, we here fo- jth atom correspondingto the upper and lower lasing levels,
2
with frequency ωj and ωj, and σj = (b a)j is the “spin- By using the above definitions of the noise operators, we
flip”operatorfortahejthatbom,with−itsad|joiihnt|σj =(a b)j. findthecorrelationfunctionsofmacroscopicnoiseforcescan
+
Thecouplingconstantgisgivenbyg = µ√ω/2~ǫ V|,iwhhe|re begenerallywrittenintheform
0
µisthemagnitudeoftheatomicdipolemoment,andV isthe
effectivevolumeofthecavity. Fk(t)Fl(t′)
In order to denote the finite-time interaction between the = (cid:10)D(0)δ(t t )(cid:11)+D(1)δ(t t τ)
kl − ′ kl − ′−
atomsandRamseyseparatedfield,weintroducethefunction + D(2)δ(t t +τ)+D(3)δ(t t τ T)
kl − ′ kl − ′− −
Γj(t)=Θ(t−tj)−Θ(t−tj−τ)+Θ(t−tj−τ−T)−Θ(t−tj−2τ−T), + D(k4l)δ(t−t′+τ+T)+D(k5l)δ(t−t′−2τ−T)
(2)
+ D(6)δ(t t +2τ+T)+D(7)δ(t t T)
whereΘ(t)istheHeavisidestepfunction[Θ(t) = 1fort > 0, kl − ′ kl − ′−
Θ(t) = 1/2 fort = 0, and Θ(t) = 0 for t < 0]. T is the free + D(k8l)δ(t−t′+T), (7)
drifttimeoftheatoms,andτistheinteractingtimebetween
theatomandonecavity. whereD(i)(k,l = a,b,M,M ;i = 0,1,2)arethequantumdif-
kl †
By the standard way [25], we can get the Heisenberg- fusioncoefficients.
Langevin equations of the motion for the single-atom and c-numbercorrelationfunctions: Bychoosingsomepartic-
filedoperators. Byintroducingthemacroscopicatomicoper- ular orderingfor productsof atomic and field operators, one
atorΓ, M(t)(σt)j=(t−),itPhejdΓyj(nta)σm−jic(te),quNaat(ito)n=sfPorjtΓhej(tfi)eσladjaa(tn)d,Nmba(ct)ro=- tchoeulqdudanertuivmetLhaencg-nevuimnbeeqrusattoiochnassdtiecriLvaendgaebvoivnee,qaunadtiaolnlsoffrothme
j j bb
Pscopicatomicoperatorsyield dynamic equations for c-number stochastic variables are the
sameasin[26].Thedifferencesarefromthecorrelationfunc-
κ
a˙(t)= a(t)+gM(t)+F (t), (3) tions. Ontheotherhand,weconvertthequantumnoiseoper-
κ
−2
atorsintothec-numbernoisevariablesF˜ (t)(k=a,b,M,M ),
k †
whosecorrelationfunctionsareexpressedas
N˙ (t)= R(1 A +A A ) (γ +γ )N (t)
a − 0 1− 2 − a a′ a
g[M (t)a(t)+a (t)M(t)]+F (t), (4) F˜ (t)F˜ (t )
† † a k k ′
−
D E
= D˜(0)δ(t t )+D˜(1)δ(t t τ)
N˙b(t)= +− gR[(aB0(t−)MB(1t)++BM2)−(t)γab(Nt)b](t+)+Fγ(a′t)N,a(t) (5) + D˜(kk2ll)δ(t−−t′′+τ)+kl D˜(k3l)−δ(t′−−t′−τ−T)
† † b + D˜(4)δ(t t +τ+T)+D˜(5)δ(t t 2τ T)
kl − ′ kl − ′− −
+ D˜(6)δ(t t +2τ+T)+D˜(7)δ(t t T)
kl − ′ kl − ′−
M˙(t)= − R(C0−C1+C2)−γabM(t) + D˜(k8l)δ(t−t′+T), (8)
+ g[N (t) N (t)]a(t)+F (t), (6)
a b M
− where D˜(i) are the c-number Langevin diffusion coefficients,
kl
wherethemacroscopicnoiseoperatorsaredefinedas related to quantum Langevin diffusion coefficients D(i) as in
kl
[27].
F (t)= Γ˙ (t)σj(t) R(1 A +A A )+ Γ (t)fj(t),
a j a − − 0 1− 2 j a Steady-state solutions: The steady-state solutions for the
Xj Xj
mean values of the field and atomic variables for laser op-
eration are obtained by dropping the noise terms of the c-
F (t)= Γ˙ (t)σj(t)+R(B B +B )+ Γ (t)fj(t), number Langevin equations and setting the time derivatives
b j b 0− 1 2 j b
Xj Xj equaltozero. Theanalyticalsolutionsareverycomplex,and
onecouldnumericallysolvethesteady-stateequations.Inthis
paper,weonlycareaboutthebadcavitylimitγ T 1
FM(t)= i Γ˙j(t)σ˜j(t)+R(C0 C1+C2) i Γj(t)fσj(t), τ 1 κ/2. Sincetheatomictransittimeismuchmasxh≪orter−tha≪n
− Xj − − − Xj th−e d≪ampingtimes of atomic variables, one could ignore the
effectof thespontaneousemissionof theatom. By the stan-
with A = σj(t +τ) , A = σj(t +τ+T) ,
0 a j q 1 a j q dardway[25],Wegetthefollowingsteady-statevalues:
A = σj(Dt +2τ+TE) , B =D σj(t +τ)E ,
2 a j q 0 b j q
CB1 == DσDbijσ(tjj(+t τ++τT) )E,q,ECB2 == Dσiσbj(jt(jDt++2ττ++TT))EEq,, A˜ss2 = R(1−A0+κ A1−A2) = R(B0−κB1+B2),
C0 = Di−σj(−t +j 2τ+EqT) .1 R is tDh−e m−eajn pumpiEnqg (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)
2 j
rate,whicDh−isd−efinedin[26]E.qItisveryeasytocheckthatthe Rτ C C +C κ
N˜ = 1+ 0− 1 2 ,
averagevaluesoftheaboveLangevinforcesareallzero. ass 2 " gτ rR(B B +B )#
0 1 2
−
3
N˜ = Rτ 1 C0−C1+C2 κ . where ΩR is the Rabi frequency on resonance,
bss 2 " − gτ rR(B B +B )# D =g2N˜ /I γ , D = g2R/2I γ2 , and
0− 1 2 ST ass 0 ab Ram 0 ab
∆ = ω (ω ω ) presents the detuning in the free
Adetailedanalysisaboutthestabilityofthesteady-statecan 2 − a2 − b2
driftregion. pisaparameter,whichcharacterizesthepump-
befoundsuchasin[28]. Inthispaper,weassumethesteady-
ingstatistics: aPoissonianexcitationstatisticscorrespondsto
statesolutionisstable.
p=0,andforaregularstatisticswehave p=1.
Laserlinwidth: Supposethe quantumfluctuationis small,
ThenthelinewidthofRamseylaserwithbadcavityisgiven
theevolutionofthefluctuationscanbeobtainedbymakinga
by
linearizationof thec-numberLangevinequationsaroundthe
steady-statesolution.Thenthemeasuredspectraoffieldfluc- γ2
D= ab D +D [2 psin2(Ω τ)sin2(∆ T)] .
tuationswillbedirectlyrelatedtothesequantities.ByFourier (κ/2+γ )2{ ST Ram − R 2 }
ab
transformationsofthelinearizedequation,we getthe ampli- (11)
tudeandphasequadraturecomponentsδX(ω)andδY(ω)[26]. SinceDST/DRam 1inoursituation,andinthecaseofmax-
Well above threshold, one can neglect the amplitude fluctu- imal photon num≪ber, the steady state value of N˜ass is about
ations, and the linewidth inside the cavity is related to the Rτ/2.Thenwegetthe
phase-diffusioncoefficient[25]. Forsmallfluctuationoflaser 2g2
phase,thespectrumofphasefluctuationsissimplyrelatedto D≈ κ [2−psin2(ΩRτ)sin2(∆2T)]. (12)
the spectrumof the phase quadraturecomponentof the field
From the expression above, we find that the pumpingstatis-
fluctuations,namely,
ticcaninfluencethelinewidth. Forregularinjection(p = 1),
1 thelinewidthisthenarrowest,whileforPoissonianinjection
(δϕ2) = (δY2) .
ω I ω (p = 0), the linewidth is the broadest. But even for regular
0
injection, the linewidth is larger than the case of one cavity.
In the region γ T 1 τ 1 κ/2, as in the recently
ab ≪ − ≪ − ≪ That means the mechanism of separated-field does not play
proposed active optical clock [15] with atomic beam. The
theroleinreducingthelinewidthasintheconventionalopti-
phase quadrature componentof the field fluctuations can be
calRamseymethod,whichiscounter-intuitive. However,the
expressedas
separatedfieldsareindispensableforanyphasedetectionlike
(δϕ2) atom interferometry. The details about the method of active
ω
atominterferometrywillappearelsewhere.
(κ/2+γ )2 g2
ab 4γ N˜ OurmethodofRamseylaserissuitableforanyatomswith
≈ I ω2[(κ/2+γ )2+ω2]4(κ/2+γ )2{ ab ass
0 ab ab metastable energy level, as an example, we choose the tran-
+ 2R[(A0+B0)+(A2+B2)] sitionfromthe metastablestate 4s4p3P to thegroundstate
1
+ Rp[(C0−C0∗)2+(C1−C1∗)2+(C2−C2∗)2]}. (9) 4s21S0of40Catocheckthestrikingfeatureofthislaser:sub-
natural linewidth. As mentioned in [29], the corresponding
SincethetimeτandT ismuchshorterthanthetimescale
naturallinewidth of the metastable state 4s4p 3P is 320Hz.
1
of the atomic dampings, we can neglect the dampings when
As in the recently proposedactive optical clock with atomic
calculateA,B,C. Byusing
i i i beam[15],thevelocityoftheatomsinthermalatomicbeamis
Ω Ω about500m/s,andthelengthoftheinteractionregionisabout
A0 =cos2 Rτ , A1 =cos2 Rτ , 1mm, then the time for the atom to traverse each coherent-
2 ! 2 !
interaction region is on the order of magnitude of 1 µs. If
a bad cavity with κ is on the order of 107Hz, the relation
A2 =1−sin2(ΩRτ)cos2 ∆22T!, B0 =sin2 Ω2Rτ!, mκ/a2gn≫ituτd−e1oifsksHatzis,fiwedh.ichThceannwbeheenasgilyisaochniethveedorfdoerrcoufrrtehnet
technique[30], from the linewidth expressionof Eq.(16)the
order of magnitude of linewidth is below 1 Hz. This means
Ω ∆ T
B1 =sin2 2Rτ!, B2 =sin2(ΩRτ)cos2 22 !, the linewidth of a Ramsey laser can be more than two or-
dersofmagnitudenarrowerthantheatomicnaturallinewidth,
thereforeourRamseymethodprovidesanewsubnaturalspec-
(C0−C0∗)2 =0,(C1−C1∗)2 =−sin2(ΩRτ)sin2(∆2T), troscopytechnique. Andsinceitisstimulated-emissionspec-
trum,itovercomesthedifficultyinothersubnaturallinewidth
(C C )2 = sin2(Ω τ)sin2(∆ T), spectroscopyschemeswherethequickreductionofsignalto
2− 2∗ − R 2 noise ratio is a formidable limit. We should point out that
weget thisRamseylaserdoesnotescapethe limitationofallactive
opticalclock: inordertopumpatomstotheexcitedstateef-
(κ/2+γ )2 γ2
(δϕ2) = ab ab D fectivelyandtobestimulatedemitphotonduringthelifetime
ω ω2[(κ/2+γab)2+ω2)](κ/2+γab)2{ ST ofametastablestate,thisnewmethodwillonlybeapplicable
+ D [2 psin2(Ω τ)sin2(∆ T)] , (10) tosomespecialtransitions[17].
Ram R 2
− }
4
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