Spectrum of spontaneous emission into the mode of a cavity QED system

0.23 MB·English

by  M. L. Terraciano

Spectrum of spontaneous emission into the mode of a cavity QED system
M. L. Terraciano,1 R. Olson,1 D. L. Freimund,1 L. A. Orozco,1 and P. R. Rice2
1Dept. of Physics, University of Maryland, College Park, MD 20742-4111, U.S.A.
2Dept. of Physics, Miami University, Oxford, OH 45056, U.S.A.
(Dated: February 1, 2008)
Westudytheprobespectrumoflightgeneratedbyspontaneousemissionintothemodeofacavity
QEDsystem. Theprobespectrumhasamaximumon-resonancewhenthenumberofinvertedatoms
for an input drive is maximal. For a larger number of atoms N, the maximum splits and develops
intoa doublet, but its frequencies are different from those of the so-called vacuumRabi splitting.
PACSnumbers: 42.50.Pq,42.50.Fx,32.80.Pj
6
0 Spontaneous emission in cavity Quantum Electrody- spontaneous emission channel in this system, together
0
namics (QED) has generated considerable interest since with the cavity output, are worth extensive study. Cav-
2
thebirthofthefield[1]. Theinteractionbetweenasingle ity QED has been identified as an environment to trans-

n atom or N atoms and a single cavity mode is different fer information and entanglement between matter and
a
from that in free space. The study of this interaction light qubits [6]. The information has to exit the sys-
J
has led to ground-breaking experiments in nonlinear op- tem through one of the two available channels as part of
0
tics, squeezing, nonclassical correlations, and quantum quantum interconnects and information protocols.
1
information [2]. Spontaneous emission in cavity QED Our cavity QED system consists of a high finesse op-

1 has been regarded as a dissipative process from which tical resonator where one or a few atoms interact with
v information is lost at a rate (γ) to modes other than the a single longitudinal and transverse mode of the cavity;
4
preferredcavitymode. MostworkincavityQEDsponta- the resulting coupling rate g depends on the dipole mo-
6
0 neous emission has focused on the enhancement or sup- ment of the transition and the electric field that carries
1 pression of the decay rate γ. An atom couples to the the energy of one photon. The combination of the rates
0 mode defined by the cavity mirrors through an allowed in this systemgives two dimensionless numbers,the first
6 transition at a rate g. Photons escape the cavity due to measures the effect of N atoms in the system through
0
/ imperfect mirror reflectivities at a rate 2κ. In the bad thecooperativityC =C1N andthesecondmeasuresthe
h cavity limit (κ >> γ,g) the resonant spontaneous emis- non-linearity of the system through the saturation pho-
t-p wsioitnhcthhaensginesglferoamtomitscforoepeesrpaatciveivtyalCue1 a=sgγ2/→(κγγ()1[+3].2CT1h)e, tinonthneumsybsetremn0a=s aγ2t/w8og-2le.vTelhaetonmon-ilnintehaeriteyxcisiteindtrsitnastiec
n
enhancement factor is related to the ratio of the atomic can not go further up the energy ladder, but only can
a
u cross section to the cavity mode cross section multiplied come downthroughspontaneous orstimulated emission;
q by the average number of reflections inside the cavity. incontrast,thecavityisaharmonicoscillatoranditsen-
: This effect broadens the spectrum, but causes no split- ergy can increase without bound. The system is driven
v
i ting [4]. There are many experimental demonstrations on-axis by a classical field ε/κ normalized normalized to
X
ofenhancedandsuppressedspontaneousemissioninthis photon flux units: y = ε/(κ√n0). Its frequency ωl is
r regime (see for example the article by Hinds in Ref. [2]). detuned from the atomic resonance ω and cavity reso-
a a
If the reflectivity of the mirrors is high enough and the nance ω by an amount Ω=ω ω , with ω =ω . The
c l c a c
−
coupling between the atom and the cavity can become atomic inversion σz is related to the expectation value
comparable to the two decays (g κ,γ), spontaneous of the intracavityfield a and the collective atomic polar-
≈
emission into the cavity is reversible. The total frac- ization σ+ with their Hermitian conjugates through the
tion of emission out of the cavity is (1 + 2C1′); where equation of motion for their expectation values:
C1′ =C12κ/(γ+2κ). The factor 2κ/(γ+2κ) is the frac-
tionofphotonsemittedintothecavitymode,exitingthe d<σz >
=<aσ+ >+<a†σ− > (<σz >+1). (1)
cavity via the mirror [5]. γdt −
Spontaneous emission plays a dual role; it is a deco- This equation shows that the atomic inversion is related
hererencesource,but it is alsoa wayto extractinforma- to the correlation between the field in the cavity and
tion out of the system. An interrogation of the system the atomic polarization. In steady state the cross terms
through spontaneous emission is an unambiguous probe <aσ+ >+< a†σ− > are proportional to the inversion.
ofthestateoftheatomicpartoftheatom-cavitysystem. Theprobespectrum(emissionasfunctionofdrivingfield
Thisletter presentsourinvestigationsofthe spectrumof frequency)isrelatedtothesteady-statemagnitudeofthe
light generated by a spontaneous emission process into cross terms of Eq. 1 as a function of driving laser laser
the mode a driven optical cavity. The properties of the frequency.
2
The system can be accurately modelled, for weak ex- 1.0
citation, as having either zero or one excitations of the units)0.8 a
coupled normal modes of the field and the atoms. If we b.
ar
assume fixed atomic positions, to first order in the ex- n (0.6
o
citation, O(y2), the equilibrium state is the pure state 0.5 missi b
[3, 7]: ns0.4
a
ψss = 0,G +x√n0 1,G p√n0 0,E +O(y4). (2) e (xy) Cavity Tr0.2
| i | i | i− | i R
2 0.0
Here n,G represents n photons with all (N) atoms in 0.0 0.5 1.0 1.5 2.0
C
| i
their ground state, n,E represents n photons with one 2
| i
atom in the excited state with the rest (N 1) in their
− C 1
ground state. The small parameter is the expectation
0
value of the intracavity field in the presence of atoms; -20 -15 -10 -5 0 5 10 15 20
Ω (MHz)
x√n0 = a . The induced atomic polarization of N
h i
atoms is p = 2Cx, which depends on the normalized
− FIG. 1: Theoretical normalized probe spectrum of sponta-
input driving field y. We can pass now from Eq.1 to the
neousemissionintothecavitymode(equivalenttotheatomic
semiclassical analysis of the weak field limit and use the inversion spectrum) (2Re(xp)). for κ/2π = 2.65 MHz and
steady state wave function to evaluate the inversion. To γ/2π =6 MHz. Inset resonant transmission for (a) the Vac-
lowestorderwehave: aσ+ = a σ+ +O(y4),whichis uum Rabispectrum and (b) atomic inversion spectra.
h i h ih i
equivalent to the decorrelation of the expectation values
of the product of the field and the polarization [8], and
we recover the Maxwell Bloch equations [9]. Figure1showsthetheoreticalcalculationofthetrans-
The spectrum of the transmitted light is given by the mitted fluorescence spectrum. This is the cross term be-
frequency dependent coefficients of the single excitation tween the field and the polarization. It starts at zero
coefficients of the steady state (Eq. 2), those of order y. for no atoms, then grows to a maximum on resonance
We canuse the state equationof cavity QEDto find the that then develops a doublet. The peaks occur at Ωxp =
transmittedspectruminboththefieldandtheinversion. g2N (κ2+(γ/2)2)/2, a different value than those
±p −
The optical bistability literature [9] gives the relation forthetransmittedspectrum,whichsplitswithpeaksat:
bizeattwioenen<thae>e,xpe<ctaσt+ion>vaanludesmoeafstuhreabfileeldquaanndtitpieoslaars- ΩX = ±qg2Np1+((γ/g2N)(γ/2+κ))−(γ/2)2 [10].
As the cooperativity grows C >>1 the peaks of the flu-
the normalized transmitted intensity, X = x2, and the
normalized incident intensity, Y = y 2. T|he| transmit- orescencespectrum into the cavity mode approachthose
| | ofthecoupledatom-cavitysystem. Themaximumonres-
ted and incident fields are related by the state equation
onance Ω=0 occurs when a given drive y excites to the
y =x(1+2C). Theatomsrespondtotheexternaldriving
upper state an optimal number of atoms, before stimu-
field by creating a polarization p = < σ+ > that op-
− latedemissiontakesovermovingthespectrumawayfrom
posesthatfieldandalmostcancelsitinthe lowintensity
the center into an Autler-Townes-like doublet.
limit. In terms of the normalized fields (assuming equal
The inset in Fig.1 shows the normalized transmitted
phases as we are treating the resonant case), the total
spectra on resonance for the transmitted intensity and
intensity in the absence of atoms (Y) in terms of the in-
the spontaneous emission. The transmitted intensity (a)
tensityinthepresenceofatoms(X)andthepolarization
starts at the peak of the transmission of the empty cav-
is:
ityanddecreasesmonotonicallywithC,whiletheatomic
Y = y 2 = x+p2 = x2+2Re(xp)+ p2 =X+F. (3) inversion (b) starts at zero, grows and has a maximum
| | | | | | | | for C = 0.5, and then decreases. The maximum coin-
The total incoming energy Y goes out as transmission cides with the place where the spectrum splits into two
X or as fluorescence F = 2Re(xp)+ p2. The fluores- peaks. The two peaks remain with a maximum value of
| |
cence has two components, one is the magnitude of the 1/2 independent of C.
polarization and the other a cross term between the in- It is difficult to experimentally study the spontaneous
tracavity field and the polarization(2Re(xp)), similar to emission in cavity QED. Work in the past has focussed
what we have in the equation of motion of the atomic on geometries that allow observation of the atoms from
inversion (Eq. 1). The intensity escaping though the theside[11]. Anotherapproachlooksatthefluorescence
cavity mode has a contribution from this term. It is into the mode of the cavity with the atoms driven by
ratherdifficulttoseparatefromthedriveandfromstim- a laser that propagates perpendicular to the cavity axis
ulated emission. However, it contains non-trivial infor- [12,13]. WefollowBirnbaumetal. [14]todirectlyaccess
mationabouttheatomicinversion,eveninthecasewhen asmallpartofthe atomic inversion. We usethe internal
<σ >= 1+O(x2),atthelowintensitylimit(x<<1). structure of the atoms to inform us when a transmit-
z
−
3
tedphotonoriginatesinafluorescenceevent(seeFig.2). from the cavity on the detector side. The cavity finesse
Instead of utilizing Rb atoms in their stretched states forthisarrangementis 21000anditsdecayconstant
F ≈
(m =F with∆m=1)toformaclosedtwo-levelsystem κ/2π=2.6 MHz. The separation between the mirrors is
F
when driven with circularly polarized light, we prepare 2 mm so the coupling coefficient between the mode and
the atoms into the m = 0 ground state and drive the thedipoletransmissionofRbisg/2π=2MHz. Themir-
F
opticaltransitionwithπ polarization(∆m=0). We can rors are glued directly to flat piezo-electric-transducers
then look at the light emitted out of the cavity separat- (PZT) to control the length of the cavity.
ingitintothetwolinearpolarizations,oneparalleltothe Ourexperimentalsystemisintheintermediateregime
drive and the other orthogonal to the drive. The pres- of cavity QED, where g (κ,γ/2). Its rates are
≈
ence of any light of orthogonal polarization signals that (g,κ,γ/2)/2π = (2,2.6,3.0) MHz. Since single atom co-
it comes originally from a spontaneous emission event of operativityC1 =0.25andthesaturationphotonnumber
an atom that decays with ∆m 1. n0 =1.1the systemrequiresaboutone photoninsteady
±
state to become non-linear, but it starts to show the ef-
B fects of spontaneous emission at a much lower intensity.
PMT WelockthecavitywithaPound-Drever-Halltechnique
usinga820nmlaser. Thislaserislockedtothestabilized
780nmlaserusingatransfercavity. Weseparatethetwo
Probe
wavelengths at the output of the physics cavity with a
grating,anduseappropriateinterferencefilterstofurther
PBS PMT
Optical ensure the separation of the two colors.
Pumping
We launch the atoms from the MOT towards the cav-
F' = 4, m' = 0 ity with a near-resonant probe beam from below. The
MOT repetition rate also sets the number of atoms delivered
tothe cavity. Wetakedatabyrecordingthetransmitted
lightinthetwoorthogonallinearpolarizationsforafixed
excitationfrequency Ω. Thereis a slightnon-degeneracy
Push Beam
F = 3, m = -1 0 1 of the two orthogonal modes of less than 0.5 MHz (less
thanthefullwidthathalfmaximumofthetransmission).
FIG. 2: Schematic of the experimental apparatus. A po- The birefringence of the cavity is less than 1 10−4 on
larizer at the output separates the two orthogonal linear po- ×
its axis.
larizations, one parallel to the driving field, the other per-
pendicular and coming from the decay through ∆m = ±1
1.0
spontaneous emission.
The apparatus (see Fig. 2) consists of two main com-
0.8
ponents: The sourceofatomsandthecavity. Twolasers ctra
provide the excitation radiation for the atomic source e
p0.6
and for the cavity. A titanium sapphire laser (Ti:Sapph) d S
provides most of the light needed for the experiment at ze
780nm. The laserlinewidth and long-termlock arecon- mali0.4
trolled using a Pound-Drever-Hall technique on satura- or
N
tion spectroscopy of 85Rb. A second laser repumps the 0.2
atoms that fall out of the cycling transition in the trap.
ArubidiumdispenserdeliversRbvaportoamagneto- 0.0
-40 -30 -20 -10 0 10 20 30 40
optical trap (MOT) in a glass cell 20 cm below a cubic Ω (MHz)
chamber that houses the cavity. The glass cell has a
silane coating to decrease the sticking of Rb to the walls FIG. 3: Transmitted intensity spectrum: vacuum Rabi in
andmaximizethecaptureefficiencyoftheMOT[15]. We filled squares; atomic inversion in empty triangles. The line
useasixbeamconfigurationwith1/ediameterof20mm shows the calculated spectrum for the spontaneous emission
intothemodeofthecavitywiththeheightadjustedtomatch
(power)and 30 mW per beam. A pair of anti-Helmholtz
thenormalized data.
coils generates a magnetic field gradient of 6 G/cm and
three sets of independent coils zero the magnetic field at The geometry that we use allows only π transitions
the trapping region. (∆m = 0) and no Faraday rotation of the light since
The cavity defines a TEM00 mode with two 7 mm di- an external uniform magnetic field is aligned with the
ameter mirrors with different transmission coefficients. polarization direction of the incoming light. The ob-
Theinputtransmission(15ppm)issmallerthantheout- served light at the orthogonal polarization must come
put (250 ppm) to ensure that most of the signal escapes from spontaneous emission. This light is emitted into
4
the cavity mode so its detection is straightforward. The ization (cavity-atom doublet) starts earlier that the one
input drive Y is polarized horizontally to better than in the orthogonal polarization (atomic inversion). For
1 10−5 andalignedtothe magneticfieldto betterthan sufficiently high number of atoms, the positions of the
×
8 degrees. two peaks coalesce into the same doublet in our simpli-
±
As each launch of atoms (every 150 ms) traverses the fiedmodel. Theoverallhorizontalscalingofthisplothas
cavity we record the transmission of both polarizations been adjusted based on the relationship between C and
(parallel and orthogonal) in a digital storage scope. We Ω , and does not take into account any other broaden-
X
then change the frequency of the driving laser and pro- ing mechanisms. Fitting C to the size of the resonant
ceed to average over 200 launches of atoms. We extract transmitted light gives consistent results within 30% to
from the raw data plots of the transmission spectrum those using Ω .
X
for a given C. Figure 3 shows the transmitted spec- The study of the spectrum of spontaneous emission
trum for the two orthogonal polarizations, the one re- into the mode of a cavity QED system shows quantita-
lated to the intracavity field (X) in filled squares (par- tivelydifferentbehaviorfromthevacuumRabispectrum.
allel polarization) and the other the one related to the The labelling of the photons by polarization permits us
spontaneousemissionin empty triangles(orthogonalpo- to identify an emissionout of the cavity generatedby an
larization). The peaks of the spectrum for X are more excited atom spontaneous decay. This will allow mea-
separated than those of the spontaneous emission. The surements conditioned on the detection of a fluorescent
calculatedspectrum forthe spontaneousemissionis nor- photon in the future. The specific quantum dynamics of
malized to match the data. this photon with orthogonal polarization remain to be
Adetailedstudyofthesusceptibilityofthisatomicsys- explored, in particular in the regime where we can no
temanditsresponsewithtwoorthogonalpolarizationsis longer neglect higher order excitations.
beyondthescopeofthisletterandwillbepresentedelse-
ThisworkwassupportedbyNSFandNIST.Wewould
where [16]. We do not make any comments here about
like to thank H. J. Kimble for his interest in this work.
the predictions of the shape of the spectrum based on
thesimplificationsofourmodel. Bothpolarizationshave
contributionsfromspontaneousemission,butonlyinthe
[1] J. J. S´anchez Mondrag´on, N. B. Narozhny, and J. H.
orthogonalpolarizationtothedrivingfieldthelightcom- Eberly, Phys. Rev.Lett. 51, 550 (1983).
ingfromthedecayofanexcitedatomisclearlyidentified. [2] Cavity Quantum Electrodynamics, Advances in Atomic,
Molecular, and Optical Physics, edited by P. R. Berman
(Academic Press, Boston, 1994), supplement 2.
15 [3] H. J. Carmichael, R. J. Brecha, and P. R. Rice, Opt.
Commun. 82, 73 (1991).
10 [4] D.J.Heinzen,J.J.Childs,J.E.Thomas,andM.S.Feld,
) Phys. Rev.Lett. 58, 1320 (1987).
z
H [5] G.CuiandM.G.raymer,Opt.Express13,9660(2005).
M 5
[6] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi,
(
y Phys. Rev.Lett. 78, 3221 (1997).
c 0
n [7] R.J.Brecha,P.R.Rice,andM.Xiao,Phys.Rev.A59,
e
u 2392 (1999).
q -5
e [8] H. J. Carmichael, An Open Systems Approach to Quan-
r
F tum Optics, Lecture Notes in Physics (Springer-Verlag,
-10
Berlin, 1993), Vol. 18.
[9] L. A. Lugiato, in Progress in Optics, edited by E. Wolf
-15 (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 69–
0 2 4 6 8 10
216.
C
[10] J. Gripp and L. A.Orozco, QuantumSemiclass. Opt. 8,
823 (1996).
FIG. 4: Evolution of the position of the doublet splitting in [11] J. J. Childs, K. An, M. S. Otteson, R. R. Desari, and
the transmitted intensity. Filled squares show the intracav- M. S.Feld, Phys. Rev.Lett. 77, 2901 (1996).
ityintensity orvacuumRabisplitting (parallel polarization); [12] Y.Zhu,A.Lezama,T.W.Mossberg,andM.Lewenstein,
empty triangles the spontaneous emission spectrum (orthog- Phys. Rev.Lett. 61, 1946 (1988).
onal polarization). The thick line is the prediction of ΩX, [13] M. Hennrich,A.Kuhn,and G. Rempe,Phys.Rev.Lett.
while thethin line corresponds toΩxp. 94, 053604 (w005).
[14] K.M.Birnbaum,A.Boca,R.Miller,A.D.Boozer,T.E.
Figure 4 presents a comparison of the position of the Northup,and H. J. Kimble, Nature 436, 87 (2005).
peaksinthemeasuredspectraforbothpolarizationswith
[15] S. Aubin, E. Gomez, L. A. Orozco, and G. D. Sprouse,
the predictions of our simple theory as function of C Rev. Sci. Instrum.74, 4342 (2003).
which in our case varies because of the change in the [16] M. L. Terraciano, R. Olson, and L. A. Orozco, To be
number of atoms. The separation of the parallel polar- published .