Relaxation oscillations, stability, and cavity feedback in a superradiant Raman laser

0.81 MB·English

by  Justin G. Bohnet

Relaxation oscillations, stability, and cavity feedback in a superradiant Raman laser
Justin G. Bohnet,∗ Zilong Chen, Joshua M. Weiner, Kevin C. Cox, and James K. Thompson
JILA, NIST and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA
Weexperimentallystudytherelaxationoscillationsandamplitudestabilitypropertiesofanoptical
laseroperatingdeepintothebad-cavityregimeusingalaser-cooled87RbRamanlaser. Bycombining
measurements of the laser light field with non-demolition measurements of the atomic populations,
we infer the response of the the gain medium represented by a collective atomic Bloch vector. The
results are qualitatively explained with a simple model. Measurements and theory are extended to
includetheeffectofintermediaterepumpingstatesontheclosed-loopstabilityoftheoscillatorand
the role of cavity-feedback on stabilizing or enhancing relaxation oscillations. This experimental
studyofthestabilityofanopticallaseroperatingdeepintothebad-cavityregimewillguidefuture
development of superradiant lasers with ultranarrow linewidths.
2
1 PACSnumbers: 42.50.Nn,42.60.Rn,42.55.Ye,42.50.Pq
0
2
p Optical lasers operating deep in the bad-cavity or and intensity fluctuations characterized by correlation
e superradiant regime, in which the cavity linewidth κ functions[16, 18].
S is much larger than the gain bandwidth γ⊥, have at- In good-cavity optical lasers, the atomic polarization
4 tractedrecenttheoretical[1,2]andexperimental[3]inter- (proportionaltoJ⊥)canbeadiabaticallyeliminatedand
est. The interest has been partially driven by the possi- therelaxationoscillationsareassociatedwiththeflowof

] bilityofcreatingspectrallynarrowlaserswithlinewidths energy back and forth between the gain inversion (pro-
s
c ≤ 1 millihertz and dramatically reduced sensitivity to portional to J ) and the cavity field A, where J , J
z z ⊥
ti the vibrations that limit state of the art narrow lasers are components of the collective Bloch vector J(cid:126) describ-
p
and keep them from operating outside the laboratory ing the atomic gain medium. In contrast, in a bad-
o
. environment[4]. Theselasersmayimprovemeasurements cavity laser, the cavity field can be adiabatically elim-
s of time[5], gravity[6], and fundamental constants[7, 8] inated, and the oscillations are driven by the coupling
c
i aidingthesearchforphysicsbeyondthestandardmodel. of J and J . Here, we will measure and infer not only
s ⊥ z
y The cold-atom Raman superradiant laser utilized here the light field A(t), but also the atomic degrees of free-
h operatesdeepintothebad-cavityregime(κ/γ⊥ ≈103 (cid:29) dom J⊥(t) and Jz(t) using non-demolition cavity-aided
p 1), making it an important physics test-bed for funda- measurements[19]togivethecompletepictureofthedy-
[ mental and practical explorations of bad-cavity optical namics of relaxation oscillations in a bad-cavity laser.
1 lasers. We will also consider the effects on the laser’s ampli-
v In the interest of fundamental science and in light tude stability of non-ideal repumping through multiple
4
of the potential applications, it is important to under- intermediatestates. Intermediaterepumpingstateswere
7
7 stand the impact of external perturbations on lasers op- notincludedinprevioussimpletheoreticalmodels[1],but
0 erating deep into the bad cavity regime. In this Let- are present in most actual realizations. In addition, we
. ter, we present an experimental study of the response to demonstratethatthecavityfrequencytuninginresponse
9
0 external perturbations of the amplitude, atomic inver- to the distribution of atomic population among various
2 sion, and atomic polarization of an optical laser operat- ground states can be used to suppress or enhance relax-
1 ing deep into the bad-cavity regime. In contrast, exper- ation oscillations. As evidence, we show stabilization of
:
v iments have extensively studied the amplitude stability Jz, J⊥, and A similar to observations of the suppression
i properties of good-cavity lasers (κ (cid:28) γ ) (See Ref. [9] of relaxation oscillations in good-cavity lasers[20]. The
X ⊥
and references therein). Previous experimental work in cavity frequency tuning mechanism is related to other
r
a the extreme bad cavity[3] and crossover regime[10] fo- applications of cavity feedback including the creation of
cused on the phase properties of the light and atomic nonlinearitiesforgeneratingspin-squeezedatomicensem-
medium. Amplitude oscillations, intensity noise, and bles [21], cavity cooling and amplification in atomic and
chaotic instabilities have been observed in gas lasers opto-mechanicalsystems[22],andthecontrolofinstabil-
operating near the cross-over regime (κ/γ ≤ 10)[11– ities in gravitational wave detectors [23].
⊥
14]. Relaxation oscillations of the field have been stud- Our experimental system consists of a quasi-steady-
ied deep into the bad cavity regime using masers[15] state Raman laser described in Fig. 1 and in Ref. [3].
in which the radiation wavelength is comparable to the The laser uses N =1×106 to 2×106 87Rb atoms as the
size of the gain medium, unlike in the present opti- gain medium. The atoms are trapped and laser cooled
cal system. Previous theoretical studies of amplitude into the Doppler-insensitive Lamb-Dicke regime (40 µK)
stability deep in the bad-cavity regime include stud- in a 1-D optical lattice at 823 nm formed by a standing
ies of relaxation oscillations[16], chaotic instabilities[17], wave in a moderate finesse F ≈ 700 optical cavity with
2
(a) (b) δ Γ III W ranges from 103 s−1 to 105 s−1, and r ranges from
∆ Γ3 Ω2 0.01 to 2. The repumping dominates all other homoge-
Repump I II nous broadening of the |↑(cid:105) to |↓(cid:105) transition such that
beams Γ
(780 nm) ω Ω W γ⊥ ≈ W/2. The inhomogenous broadening of the tran-
cav d sition is γ ≈ 103 s−1. To summarize, the relevant
Ω in
γ
1 rates to characterize our system satisfy the hierarchy
κ (cid:29) γ ≈ W/2 ∼ NCγ > γ (cid:29) γ. Coupling to other
⊥ in
Dressing Trap
beam (823nm) ↑ 3 transverse and longitudinal cavity modes is negligible.
(795 nm) Thefrequencyofthesuperradiantlyemittedlightω is
↓ γ
set by the frequency of the dressing laser and the hyper-
(c)
2At|()| 9-1 s][10 fiannnodremscpaallviitzitteiydngrteoωsHothnFea/n2cπcaev=iftry6e.qh8u3ae4lnfGcliynHeizws.iδTdth=he.d2Te(tωhucenavisni−nggoωlefγlp)ig/ahκrt-,
ticlescatteringratefromthedressinglaserintothecavity
mode is Γ (δ)=Cγ/(cid:0)1+δ2(cid:1).
c
The cavity frequency is dispersively tuned by the
FIG.1. (coloronline)(a),(b)Physicalsetupandenergylevel (cid:80)
atomicensembleω =ω + α N ,whereω is
diagram. The trapping light (orange) and Raman dressing cav bcav k k k bcav
the bare cavity frequency and α is the cavity frequency
laser (red, power ∝ Ω2) are injected along the cavity axis. k
d shift for a single atom in the kth ground Zeeman state
Therepumpinglight(purple,green)isappliedperpendicular
to the cavity axis. The emitted optical laser light (blue) is resulting from dispersive phase shifts of the intracavity
nearly resonant with the cavity mode (dashed lines) detuned light field.
from ωcav by δ. The repumping is accomplished through Since αk (cid:54)= αj for k (cid:54)= j, the cavity frequency can
a pair of two-photon transitions through intermediate opti-
provide a measurement of the populations N . We can
cally excited states |II(cid:105) and |III(cid:105) with incoherent decay rates k
suddenly switch off the repumping and dressing lasers
Γ. We individually control the two-photon rates W and Γ
3
withtherepumpinglaserpowers∝Ω2 . |3(cid:105)representsother to effectively freeze the atomic populations[25]. We
1,2
metastablegroundstatesbesidesthelaserlevels. (c)Example then combine repeated non-demolition cavity frequency
emittedlaserphotonflux|A(t)|2 versustimeshowingspiking measurements[19, 26, 27] and NMR-like rotations[28] to
and relaxation oscillations at turn-on. determine J (t) = (cid:104)1(cid:80)N(|↑ (cid:105)(cid:104)↑ |−|↓ (cid:105)(cid:104)↓ |)(cid:105) and δ(t)
z 2 i i i i i
[24]. The amplitude of the light field emitted from the
cavity A(t) just prior to freezing the system, along with
a cavity power decay rate κ/2π = 11 MHz. The atom-
themeasurementofthecavityfrequencydetuningδ pro-
cavity coupling to the relevant TEM00 cavity mode on videsaninferredvalueofJ =(cid:12)(cid:12)(cid:68)Jˆ (cid:69)(cid:12)(cid:12)usingtherelation
resonance with the emitted light is characterized by the ⊥ (cid:12) − (cid:12)
single particle cooperativity parameter of cavity-QED A(t)=(cid:112)Γ (δ(t))J (t), where Jˆ =(cid:80)N|↓ (cid:105)(cid:104)↑ |.
c ⊥ − i i i
C =8×10−3 (cid:28)1. We observe characteristic laser spiking and relaxation
Fig. 1b shows a simplified energy level diagram oscillation behavior in |A(t)|2 as the laser turns on and
of a three level Raman laser system. The las- settlestosteadystate(Fig. 1c). Tosystematicallystudy
ing transition is a spontaneous optical Raman tran- small amplitude deviations about the steady-state val-
(cid:12)(cid:12)s5it2ioSn1/2frFom=1|↑,(cid:105)mF≡=(cid:12)(cid:12)05(cid:11)2Sin1d/2ucFed=b2y,maFdr=es0si(cid:11)ngtolas|e↓r(cid:105)d≡e- iuneqsuoef, Jsizm,si,laArst,oanRdef.J⊥[,2s,9]w. eWapepalyppalyswaespitmsuilnteanteeocuhs-
tunedfromthe|↑(cid:105)to|I(cid:105)≡(cid:12)(cid:12)52P1/2F(cid:48) =2(cid:11)transitionby small amplitude modulation of the repumping rates as
∆/2π = +1.1 GHz. The incoherent repumping process W(t) = W(1+(cid:15)cosωt)) and Γ (t) = rW(t). The mod-
3
takes place through a third metastable ground state |3(cid:105) ulation frequency ω is scanned over frequencies of order
which represents the sum of the other hyperfine ground W, such that γ < ω (cid:28) κ. We then measure and infer
statesin87Rb. Thefullenergyleveldiagramwithdetails thequantitiesA(t),J (t),andJ (t)asdescribedearlier.
⊥ z
of the dressing and repumping lasers is provided in Ref. To calculate the transfer function of the applied mod-
[24]. ulation, the measured light field amplitude A(t) exit-
We control the dynamics of the system primarily ing the cavity as a function of time is fit to A(t) =
through three two-photon population transfer rates: γ, A (1+a(ω)cos(ωt+φ (ω))). The normalized fractional
s a
W, and Γ3. We control γ (typical value 60 s−1) using amplitude response transfer function is TA(ω) ≡ a(ω)/(cid:15)
theintensityofthe795nmdressinglaser. Wecontrolthe and the phase response transfer function is T ≡ φ (ω).
φ a
repumpingratesW andΓ3 usingtwo780nmrepumping Wealsodefinethemodulationfrequencythatmaximizes
laserstunednearresonancewiththe(cid:12)(cid:12)52S1/2F =1,2(cid:11)→ TA(ω) as the resonance frequency ωres. We present the
(cid:12)(cid:12)52P3/2F(cid:48) =2(cid:11) transitions. The repumping intensities results in Figs. 2, 3, and 4, which we discuss in detail
are independently controlled allowing us to set the pro- after developing a theoretical model to guide the inter-
portionality factor r ≡ Γ /W. In our experiments, petation.
3
3
(a) ω/2π = 0.11 kHz ω/2π = 2.2 kHz ω/2π = 8 kHz at W = Wpk = 21NΓc(δs) where δs is the steady state
cavity detuning.
To predict relaxation oscillations and damping, we
60) do a straightforward expansion about the steady state
1
J(z values Jz,s and J⊥,s as Jz(t) ≈ Jz,s(1 + jz(t)) and
J (t)≈J (1+j (t))andignoretermsthataresecond
⊥ ⊥,s ⊥
2ε = 0.22 ε = 0.11 2ε = 0.22
order in small quantities j ,j and repumping modula-
⊥ z
tion amplitude (cid:15)[16, 32]. This results in a system of cou-
J (106) J (106) J (106)
⊥ ⊥ ⊥ pled differential equations for j ,j which can be recast
z ⊥
(b)
as two uncoupled higher order differential equations
β·j·· +¨j +2γ j˙ +ω2j =D (ω)(cid:15)eiωt (1)
z,⊥ z,⊥ 0 z,⊥ 0 z,⊥ z,⊥
which, ignoring the third order term, is similar to the
π equation for a damped harmonic oscillator described by
a damping rate γ , a natural frequency ω , and a com-
π/2 0 0
Hz]2 plex, frequencydependentdriveDz,⊥(ω). Todiscussthe
-π/20 ω [kres sζta=biγli0ty. Aofttchreitsicyasltedma,mwpeinaglsζo=defi1n,entohreedlaaxmatpiionngorsactiilo-
0 0.4 0.8 ω0
W [W ] lations are present, as the system exponentially relaxes
-π max
to steady state.
We will consider relevant limiting cases of these equa-
tions to concisely guide the interpretation of the data.
FIG.2. (coloronline)(a)ParametricplotsoftheBlochvector The full equations and expressions for γ0, ω0, β and
componentsJz(t)andJ⊥(t)overasinglecycleofmodulation Dz,⊥(ω) are given in Ref. [30].
of the repumping rate W near Wpk for modulation frequen- First we consider the limiting case where r → ∞ and
cies below, near, and above resonance or ω/2π = 0.11,2.2,8
δ = 0 for which J and J respond like a damped
z ⊥
kHz, from left to right. The black points are the mea-
harmonic oscillator described by γ = W/2 and ω =
suredsmall-signaldeviationsaboutthemeasuredsteady-state (cid:112) 0 0
W(NCγ−W) as β = 0. However, the drive on
Bloch vector (red arrow). The blue curve is the predicted
deviation from steady-state given the experimental parame- the two oscillators is different, leading to different over-
ters N = 1.3×106, r = 0.71, δs = 1, W = 0.4NCγ, and all responses as D⊥(ω) = W2 (NCγ − 2W − iω) and
NCγ = 125×103s−1. The modulation depth (cid:15) for data be- D (ω)=(NCγ−W)(W+iω). Themagnitudeandphase
z
low and above resonance was doubled to make the response of the drives change with the modulation frequency and
more visible. (b) The amplitude (upper) and phase (lower)
repumping rate, even as the modulation depth (cid:15) remains
response transfer functions T (ω),T (ω) of the light field for
A φ constant.
three values of the repumping rate W. The points are mea-
WeshowthisdrivenoscillatorresponseinFig. 2awith
sured data, and the lines are zero free parameter predictions
of the response. (inset) ωres versus W (points) and a fit to themeasuredandpredictedparametricplotofJz andJ⊥
ω (line) showing the expected frequency dependence of the at three different applied modulation frequencies, with
0
relaxation oscillations on repumping rate. repumping near W = W . Although the characteris-
pk
tic frequencies and rates of the atomic oscillator do not
change, thedifferingdrivesleadtoachangeinthephase
relationshipbetweentheresponseofthetwoquadratures.
The theoretical model presented here consists of equa-
This data demonstrates how the external perturbation
tions for the three ground state levels shown in Fig. 1b,
results in different effective drives for j and j , leading
which qualitiatively describes the eight level 87Rb sys- ⊥ z
to different behaviors with modulation frequency.
tem. The data is also quantitatively compared to an
In Fig. 2b, we focus on the light field’s transfer func-
eight-level model described in Ref. [30]. Our theoreti-
tions T (ω) and T (ω). Data for three different average
cal model uses the semi-classical optical Bloch equations A φ
repumping rates W are shown. The data qualitatively
for an effective two level system where κ(cid:29)W,γ so that
agrees with Eqn. 1, and quantitatively agrees with the
the cavity field is effectively slaved to the atomic po-
displayed theory calculated for the full model including
larization and can be adiabatically eliminated from the
all 87Rb levels.
system of equations. Additionally, we have adiabati-
In most experimental systems for a superradiant light
cally eliminated the populations in the optically excited
source, the repumping ratio r is finite. In this case,
states|I(cid:105),|II(cid:105),|III(cid:105),arrivingatthesteadystatesolutions
the damping rate and resonance frequencies are γ =
fJo⊥r,st[h1,e3i1n]v.eTrshioenstJeza,dsyasntdatecoallmecptliivteueaAtosmiiscmcaoxhiemreiznecde r(W2((12+rr−)1(1)++N2rC)γ) and ω0 =(cid:113)1+rrW(NCγ−W). We0can
4
(a) (a)
(b)
(b)
FIG.3. (coloronline)Effectsoffiniteratioofrepumpingrate
r (a) Comparison at two values of r of T (ω)versus modula-
A
tion frequency. The points are measured data in good agree-
ment with the zero-free parameter fit (lines). (b) Plot of the
resonancefrequencyω (black)andthepeakvalueT (ω )
res A res
FIG.4. (coloronline)Evidenceofnegativeandpositivecavity
(red) versus r.
feedback. (a) Amplitude transfer functions of the emitted
electric field for three detunings from cavity resonance. The
datapointsaretheaverageof4experimentaltrials. Thelines
understand the effect of the nonzero β = 1 on the are fitted transfer functions with N as a free parameter. (b)
W(1+r)
response by considering the Fourier transform of Eqn. Cavitydamping of collectiveatomic degreesof freedom. The
response going from δ = 0.2 (left) where the γ is small to
1. The effective damping rate is the coefficient on the s 0
δ =0.9 (right) where the system is expected to be critically
sumoftheimaginarytermsonthelefthandsideofEqn. s
damped. Theredlinesaresinusoidalfitstothedata(circles).
1, γ(cid:48) = γ −ω2/(W(1+r)). The frequency dependent
0 0 The dashed lines highlight the damping in both J⊥ and Jz.
termassociatedwithβ lowerstheeffectivedampingrate.
However, as long as γ(cid:48) > 0 at ω = ω , the system will
0 0
remain stable. The form of γ(cid:48) also indicates reduced bothatomicdegreesoffreedomarestabilizedbynegative
0
damping for r < 1 as shown in Fig. 3a. We observe an cavity-feedback.
increase in the peak relaxation oscillation amplitude as We have studied the dynamics of the polarization, in-
r → 0 (red). From the form of ω0, we also expect to version,andfieldofanopticallaseroperatingdeepinthe
see the resonance frequency sharply decrease as r → 0, bad-cavity regime. We have shown that dispersive cav-
which we observe in Fig. 3b (black). ity frequency tuning can suppress or enhance relaxation
To understand the dynamic tuning of the cavity reso- oscillations. Having experimentally validated our model
nancefrequencyω inresponsetochangesintheatomic foropticallasersintheextremebad-cavityregime,future
cav
populations,wenextconsiderthecaser →∞andδ (cid:54)=0. work can now extend the formalism to realistic models
We also assume the cavity frequency is tuned by the of proposed ultrastable lasers using ultranarrow atomic
atomic inversion J as α = α = −α > 0. The dy- transitions in atoms such as Sr and Yb[1]. Further stud-
z ↓ ↑
namic cavity tuning then modifies the damping rate as ies of the nonlinear dynamics of the extreme bad-cavity
(cid:16) (cid:16) (cid:17)(cid:17)
γ = W 1+2αδ N − W . The tuning of ω laser system will include investigations of chaos[17] and
0 2 s 1+δ2 Cγ cav
s squeezed light generation[33].
can act as either positive or negative feedback. When
The authors acknowledge helpful discussions with D.
δ > 0 the additional term enhances the damping, mak-
s
Meiser, M. J. Holland, J. Ye, D. Z. Anderson, and S.
ingtheoscillatormorestable. Ifδ <0,thedampingcan
s
T. Cundiff. This work was funded by NSF PFC, NIST,
become negative, leading to an unstable oscillator.
ARO, and DARPA QuASAR. J.G.B. acknowledges sup-
Weobservethisqualitativebehaviorinthedata,which
port from NSF GRF, and Z.C. acknowledges support
agreeswellwithafullmodel[30]. Asδ isvariedfrom<0
s from A*STAR Singapore.
to > 0, the cavity feedback first creates positive feed-
back that enhances and then negative feedback that de-
creasesthemeasuredamplituderesponsefunctionT (ω)
A
in Fig. 4a. We also measure the effect of the damp-
ing on J (t) and J (t) in Fig. 4b, where we can see that ∗ Send correspondence to: [email protected]
z ⊥
5
[1] D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, [29] J. K. Wahlstrand, J. T. Willits, T. R. Schibli, C. R.
Phys. Rev. Lett. 102, 163601 (2009). Menyuk, andS.T.Cundiff,Opt.Lett.32,3426(2007).
[2] J. Chen, Chinese Science Bulletin 54, 348 (2009). [30] “Manuscript in preparation”.
[3] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. [31] D. Meiser and M. J. Holland, Phys. Rev. A 81, 033847
Holland, and J. K. Thompson, Nature 484, 78 (2012). (2010).
[4] Y.Y.Jiang,A.D.Ludlow,N.D.Lemke,R.W.Fox,J.A. [32] A.E.Siegman,Lasers,1sted.(UniversityScienceBooks,
Sherman, L.-S. Ma, and C. W. Oates, Nature Photon. 1986).
5, 158 (2011). [33] A. Lambrecht, J. M. Courty, S. Reynaud, and E. Gi-
[5] A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, acobino, Applied Physics B: Lasers and Optics 60, 129
M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. (1995).
Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E.
Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber,
N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates,
Science 319, 1805 (2008).
[6] C. W. Chou, D. B. Hume, T. Rosenband, and D. J.
Wineland, Science 329, 1630 (2010).
[7] T. M. Fortier, N. Ashby, J. C. Bergquist, M. J. Delaney,
S.A.Diddams,T.P.Heavner,L.Hollberg,W.M.Itano,
S. R. Jefferts, K. Kim, F. Levi, L. Lorini, W. H. Oskay,
T.E.Parker,J.Shirley, andJ.E.Stalnaker,Phys.Rev.
Lett. 98, 070801 (2007).
[8] S.Blatt,A.D.Ludlow,G.K.Campbell,J.W.Thomsen,
T.Zelevinsky,M.M.Boyd,J.Ye,X.Baillard,M.Fouch´e,
R.LeTargat,A.Brusch,P.Lemonde,M.Takamoto,F.-
L. Hong, H. Katori, and V. V. Flambaum, Phys. Rev.
Lett. 100, 140801 (2008).
[9] D. E. McCumber, Phys. Rev. 141, 306 (1966).
[10] S.J.M.Kuppens,M.P.vanExter, andJ.P.Woerdman,
Phys. Rev. Lett. 72, 3815 (1994).
[11] L. Casperson, IEEE J. of Quantum Electron. 14, 756
(1978).
[12] M. A. van Eijkelenborg, M. P. van Exter, and J. P.
Woerdman, Phys. Rev. A 57, 571 (1998).
[13] R.G.HarrisonandD.J.Biswas,Phys.Rev.Lett.55,63
(1985).
[14] C.O.Weiss,N.B.Abraham, andU.Hu¨bner,Phys.Rev.
Lett. 61, 1587 (1988).
[15] J.H.Shirley,AmericanJournalofPhysics36,949(1968).
[16] M. I. Kolobov, L. Davidovich, E. Giacobino, and
C. Fabre, Phys. Rev. A 47, 1431 (1993).
[17] H. Haken, Physics Letters A 53, 77 (1975).
[18] D. Meiser and M. J. Holland, Phys. Rev. A 81, 063827
(2010).
[19] Z. Chen, J. G. Bohnet, S. R. Sankar, J. Dai, and J. K.
Thompson, Phys. Rev. Lett. 106, 133601 (2011).
[20] H. Statz, G.A. DeMars, D.T. Wilson, and C.L.Tang,
Journal of Applied Physics 36, 1510 (1965).
[21] I.D.Leroux,M.H.Schleier-Smith, andV.Vuleti´c,Phys.
Rev. Lett. 104, 073602 (2010).
[22] T. J. Kippenberg and K. J. Vahala, Science 321, 1172
(2008).
[23] L.Ju,D.G.Blair,C.Zhao,S.Gras,Z.Zhang,P.Barriga,
H.Miao,Y.Fan, andL.Merrill,ClassicalandQuantum
Gravity 26, 015002 (2009).
[24] “See supplemental material at [url will be inserted by
publisher]”.
[25] J. G. Bohnet, Z. Chen, J. W. Weiner, K. C. Cox, and
J. K. Thompson, arXiv:1208.1710v2 [physics.atom-ph].
[26] I. Teper, G. Vrijsen, J. Lee, and M. A. Kasevich, Phys.
Rev. A 78, 051803 (2008).
[27] M.H.Schleier-Smith,I.D.Leroux, andV.Vuleti´c,Phys.
Rev. Lett. 104, 073604 (2010).
[28] Z.Chen,J.G.Bohnet,J.M.Weiner, andJ.K.Thomp-
son, Review of Scientific Instruments 83, 044701 (2012).
Relaxation oscillations, stability, and cavity feedback in a superradiant Raman laser:
supporting online material
Justin G. Bohnet, Zilong Chen, Joshua M. Weiner, Kevin C. Cox, and James K. Thompson
JILA, NIST and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA
PACSnumbers: 42.50.Nn,42.60.Rn,42.55.Ye,42.50.Pq
(a) Raman Transition (b) Repumping Transitions gram shown in Fig. 5). The |↑(cid:105) state is dressed with
D1, 795 nm D2, 780 nm an optical Raman laser at 795 nm that induces an op-
F'2 tical decay to |↓(cid:105), with the cavity tuned to be resonant
or near-resonant with the emitted light. The intensity
of the dressing beam and the detuning by ∆/2π =+1.1
I F'2 GHzfromtheopticallyexcitedintermediatestates|I±(cid:105)≡
(cid:12)(cid:12)52P1/2F(cid:48) =2,m(cid:48)F =±1(cid:11) set the single-atom scattering
rate into all of free space γ. The induced single-atom
scattering rate into the cavity mode is Γ = Cγ where
c 1+δ2
δ is the detuning of the cavity resonance frequency from
e F2 F2 the emitted light in units of the cavity half-linewidth,
and C is the single particle cooperativity parameter of
cavity QED[2]. The branching ratio for decay to |↓(cid:105) are
g F1 F1 included in the definition of C.
mf = -1 0 1 mf = -2 -1 0 1 2 The quantization axis is set by a 2.7 G magnetic field
along the cavity axis. The Raman dressing laser is in-
FIG.5. (a)TheenergyleveldiagramfortheD1Ramantran- jectednon-resonantlyalongthecavityaxis(experimental
sitionusedforlasing. ThelinearlypolarizedRamandressing
setup shown in Fig. 6a). For this quantization axis, the
laser is shown in red and the superradiantly emitted light in
linearly polarized Raman dressing laser is an equal com-
blue. The cavity mode resonance frequency ω is denoted
cav bination of σ+ and σ− light. Constructive interference
with a blue dashed line. With the quantization axis defined
by a 2.7 G magnetic field oriented along the cavity axis, the between the two decay paths from |↑(cid:105) to |↓(cid:105) through the
linearly polarized light is a linear combination of σ and σ two states |I (cid:105) leads to enhancement of light emission
+ − ±
polarizations. (b) The energy level diagram for the D2 re- with linear polarization rotated 90◦ from the polariza-
pumping beams F2 (green) and F1 (purple). The dark state tion of the dressing laser. Conversely, the orthogonal
with respect to the repumping lasers is labeled with a gray
polarizationofemittedlightissuppressedbydestructive
circle and corresponds to |↑(cid:105).
interference of the two decay paths.
Two repumping lasers control the rate out of |↓(cid:105)
Cavity details, full energy level diagram, dressing, and back into |↑(cid:105). The laser powers are independently
and repumping laser scheme. set using acoustic optic modulators (AOMs) 1 and 2
shown in Fig. 6a. Both repumpers are π-polarized, ap-
The optical cavity has a mode waist of 71 µm, mirror plied perpendicular to the cavity, and separately tuned
separationof1.9cm,andafinesseofF =700. Theatoms near the (cid:12)(cid:12)52S1/2F =1,2(cid:11) to (cid:12)(cid:12)52P3/2,F(cid:48) =2(cid:11) transi-
are trapped by a 1-D intracavity optical lattice at 823 tions. The F1 repumper moves atoms primarily from
nm and laser cooled to approximately 40 µK. The sub- the ground (cid:12)(cid:12)52S1/2F =1,mF(cid:11) states to the ground
wavelengthlocalizationoftheatomsalongthecavity-axis (cid:12)(cid:12)52S1/2F =2,mF(cid:11) states, and sets the scattering rate
ensures that the atoms are in the Lamb-Dicke regime Wout of |↓(cid:105). The F2 repumper pushes population
along this direction. However, the atoms are not in the to |↑(cid:105) as the dipole matrix element for the transition
Lamb-Dicke regime with respect to motion transverse to |52S ,F = 2,m = 0(cid:105) → |52P ,F(cid:48) = 2,m(cid:48) = 0(cid:105)
1/2 f 3/2 f
the cavity axis. is zero. We quantify the F2 repumping rate by calcu-
We first note that the energy levels, dressing laser, lating the single particle scattering rate for an atom in
repumping lasers, and relative frequency tunings are |52S ,F =2,m =1(cid:105)stateasΓ . Wedefinethepump
1/2 f 3
the same as the primary configuration presented in ratio r ≡ Γ3 to quantify to what degree population can
W
Ref. [1]. We include the details here again for clar- build up in energy levels outside of |↑(cid:105) and |↓(cid:105), with the
ity. The |↓(cid:105) ≡ (cid:12)(cid:12)52S1/2F =1,mF =0(cid:11) and |↑(cid:105) ≡ ideal repumping case being r →∞. Population will still
(cid:12)(cid:12)52S1/2F =2,mF =0(cid:11) hyperfine ground states of 87Rb remain in the states |52S1/2,F = 1,mf = ±1(cid:105) even as
form the basis of our Raman laser (energy level dia- r →∞, unlike in the ideal three-level model. This is the
7
(a) ADC W (b)
Repumper
Cavity
Probe I ADC J , J
Raman Q z ⊥ Dressing Laser
Dressing
Loop Local Oscillator π
DAC Filter Microwave Source
6.834 GHz
M1
Repump AO AOM Master SµoWuarvcee Cavity Frequency M1 M2
Ratio r M Probe
O
A2 t1 t2 t3 Time
FIG. 6. Measurement setup and sequence timing diagram. (a) The physical setup for measuring the light field amplitude
response A(t) and the atomic responses J (t), J to a small modulation of the repumping rate W(t). Optical beams are
z ⊥
colored with arrows; RF and microwave signals are in black. (b) Timing diagram for measurements. Superradiance is started
at t , and the repumper W(t) is modulated (with r constant) starting at time t . At t , the repumper and dressing lasers
1 2 3
are shut off, freezing the atomic populations. The microwave π-pulse corresponds to a 12 µs pulse of 6.834 GHz microwaves
resonant only with the ground hyperfine state transition |↑(cid:105) to |↓(cid:105). The pulse completely swaps the populations N and N of
↑ ↓
states|↑(cid:105)and|↓(cid:105). ThecavityfrequencyprobewindowsM1andM2usedtodeterminethedressedcavityfrequencyω and
cav,1
ω , and from which we can determine J (t ).
cav,2 z 3
reason that the output power correction factor R does tuningofthecavityresonanceperatomineachstateα
↑,↓
not asymptote to 1 as r →∞. Also note that unlike the is calculated from the known cavity geometry, atomic
simplethree-levelmodel,here,W includesRayleighscat- dipole moments, and bare cavity detuning from reso-
teringbackto|↓(cid:105),contributingdampingofthecoherence nance with nearby optical atomic transitions at 795 nm.
J without affecting the inversion J . The process is repeated for different stopping times t in
⊥ z 3
order to sample the modulation period for a set of times
t . The data J (t ) versus J (t ) are shown in several
i z i ⊥ i
Response function measurement sequence parametric plots in the main text.
In the case of the field transfer functions T (ω) and
A
In the main text, we measure three quantities to char- T (ω),continuoustimetracesA(t)areaveragedoversev-
φ
acterize the response of the superradiant laser to small eraltrialsatthesamemodulationfrequencyandphaseto
perturbations: the light field amplitude A(t), the mag- enhance signal to noise. The average response is then fit
nitude of the coherence J (t), and the inversion J (t). toA(t)=A (1+a(ω)cos(ωt+φ (ω)),withtransferfunc-
⊥ z s A
As shown in Fig. 6a and b, the perturbation is cre- tions calculated as T (ω) = a(ω)/(cid:15) and T (ω) = φ (ω).
A φ A
ated by a small fractional modulation of the F1 and The measurement is then repeated for different ω.
F2 repumping laser power at frequency ω such that
W(t) = W(1+(cid:15)cosωt) while r(t) remains constant. At
time t , we shut off the repumping and dressing lasers in
3
less than 100 ns, freezing the atomic populations. The
∗ Send correspondence to: [email protected]
amplitude of the light just before shut off A(t ) is deter-
3 [1] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J.
minedfromtheIQ-demodulatedheterodynesignalshown
Holland, and J. K. Thompson, Nature 484, 78 (2012).
in Fig. 6a.
[2] Z. Chen, J. G. Bohnet, S. R. Sankar, J. Dai, and J. K.
We then use methods similar to those in [2] to non- Thompson, Phys. Rev. Lett. 106, 133601 (2011).
destructively probe the cavity-mode to determine the
cavity resonance frequency ω and hence detuning
cav,1
from the emitted light frequency δ(t ). We then cal-
3
culate the coherence J (t ) using the relation A(t ) =
⊥ 3 3
(cid:112)
J (t ) Cγ/(1+δ2(t )). The probe is non-destructive
⊥ 3 3
in that a small fraction of the atoms are lost or Raman
scattered to other states during the measurement.
The inversion J (t ) is determined by using a mi-
z 3
crowaveπ-pulsetoswapthepopulationsbetween|↑(cid:105)and
|↓(cid:105), and then measuring the cavity resonance frequency
a second time ω . Shifts in the cavity frequency due
cav,2
to atoms in other states are common mode to both mea-
surements, suchthatthedifferencebetweenthetwocav-
ity frequency measurements is only proportional to the
inversion ω −ω =2(α −α )J . The frequency
cav,1 cav,2 ↑ ↓ z