Stabilizing an atom laser using spatially selective pumping and feedback

0.79 MB·English

by  Mattias Johnsson

Stabilizing an atom laser using spatially selective pumping and feedback
Mattias Johnsson, Simon Haine and Joseph J. Hope
ARC Centre for Quantum-Atom Optics, Faculty of Science,
The Australian National University, Canberra, ACT 0200, Australia
(Dated: February 1, 2008)
Weperformacomprehensivestudyofstabilityofapumpedatomlaserinthepresenceofpumping,
damping and outcoupling. We also introduce a realistic feedback scheme to improve stability by
extracting energy from the condensate and determine its effectiveness. We find that while the
feedbackschemeishighlyefficient inreducingcondensatefluctuations,itusually doesnotalterthe
stability class of a particular set of pumping,damping and outcoupling parameters.
PACSnumbers: 03.75.Pp,03.65.Sq,05.45.-a
5
0
I. INTRODUCTION lowertrapmodes,andideallythegroundstate. Stability
0
2 can also be improved if a spatially dependent pumping
n The experimentalrealizationofBose-Einsteinconden- scheme is utilized, where the condensate is preferentially
a sates(BECs)hasprovidedatestbedformanyfundamen- pumped in a narrow range near the center of the trap
J talissuesininteractingquantumsystems,aswellaspro- [12]. As lower frequency modes have a greater overlap
9 viding a general tool for investigating aspects of atomic with the central portion of the trap, these modes are
1 physics such as the behaviour of weakly interacting al- moreeffectivelypopulatedthanhigherfrequencymodes,
kali gases [1]. One major application that BECs offer is again leading to the promotion of low-frequency modes
1 the possibility of creating an atom laser [2]. Just as the and encouraging stability.
v
optical laser revolutionisedoptics by offering spatial and Although interatomic interactions can lead to single
1
0 temporal coherence, high spectral density and mode se- mode operation of an atom laser, strong interactions
1 lectivity, the pumped atom laser offers the possibility of will ultimately limit the linewidth, as they will cause
1 doing the same for atomic physics. phase diffusion of the lasing mode [13, 14]. An indepen-
0 Atom lasers have been achieved by outcoupling atoms dent method of improving modal stability without using
5 from a BEC using some external means to change the high interactionenergies is to use a feedback mechanism
0 state of a subset of the atoms in the condensate from whereby energy is removed from the condensate using
/
h a trapped to an antitrapped state [3, 4, 5]. This can continuous knowledge of the atomic cloud’s dynamics to
p produce a beam of atoms that exhibits both spatial and tune the trap parameters. It has been shown that if
t- temporal coherence [6, 7]. As in optical lasers, a narrow one considers an isolated condensate in a trap, with no
n linewidth (i.e. small momentum spread)canbe attained pumping, loss or outcoupling present, then provided the
a byensuringtheoutcouplingisweak,althoughthisresults quantities hxi, hx2i and h|ψ|2i can be measured, where
u ∗
q inalowbeamflux[8]. Inordertocreateahighflux,nar- hqi = ψ qψdx, then it is almost always possible to
: row linewidth atomic beam, there must be competition extractRenergyfromthe condensate,andthus drawitto-
v between a depletable pumping mechanism and damping wards the lowestenergy groundstate [15]. It is possible,
i
X resulting in gain narrowing [7]. however,thatsomespecific(non-ground)statemayexist
thatprovidesnoerrorsignaltothefeedbackloopcausing
r A multimode analysis of an atom laser is possible us-
a ing semiclassicaltechniques such as the Gross-Pitaevskii the feedbackprocesstohavenoeffect. Thisoccurswhen
all the moments chosen as error signals are stationary,
equation [1, 9]. Such a model can describe spatial varia-
i.e. do not change with time.
tion of the modes and determine if the laser approaches
single-mode operation. It cannot, however,calculate the The purpose of this paper is to tie these threads to-
linewidth of the laser in this limit, since the linewidth of gether into a comprehensive analysis of atom laser sta-
asingle-modelaserisgovernedbythequantumstatistics bility, incorporating loss, outcoupling, spatially depen-
of the mode, so that a full quantum analysis is required dent pumping, a depletable reservoir and feedback. We
[10]. Thisdrasticallyincreasesthedifficulty ofthecalcu- will consider stability across a broad range of pumping
lation, limiting current efforts to only a few modes. regimes, and include the effects of the outcoupled beam
on the condensate as this modifies some of the stability
In order for an atom laser to reach single mode op-
conclusions in Ref. [12].
eration it is necessary that it is stable, such that the
dynamics of the condensate has a steady-state solution.
It has previously been shown that if the pumping of the
II. MODEL
condensate is spatially uniform, then an atom laser will
beunstableifthenonlinearinteractionstrengthbetween
the atoms is below some critical value [11]. This is be- As discussed in the introduction, determining the sta-
cause increasing nonlinearity causes greater damping of bility of an atom laser does not require a full quantum
higherfrequencytrapmodes,leadingtothepromotionof analysis. We will therefore use a multimode semiclassi-
2
cal model based on the Gross-Pitaevskii equation. We be the position of the trap minimum potential, the trap
denote the condensate field by ψ (x) and the untrapped strength, and the nonlinear interaction strength. Conse-
t
field by ψ (x). ψ (x) forms the lasing mode, and is quently we assume it is possible to control an external
u t
pumped by anincoherentreservoirof atoms which has a potential of the form V1 = a (t)x+a (t)x2 and a non-
fb 1 2
densitydescribedbyn(x),withthecouplingbetweenthe linear interaction strength of the form V2 = b(t)|ψ |2.
two given by κ(x). For calculational tractability, we re- The a (t), a (t) and b(t) correspond to tfimbe-dependetnt
1 2
strict ourselves to a one dimensional condensate, so that control parameters that can be manipulated according
the dynamical equations for the fields can be written as to the measured error signals. Altering a and a cor-
1 2
dψ ¯h2 responds to changing the position of the trap minimum
i¯h t = − ∇2+V −i¯hγ(1)+ U −i¯hγ(2) |ψ |2 and its curvature, while tuning b(t) corresponds to ma-
dt (cid:20) 2m t t (cid:16) tt t (cid:17) t nipulatingthenonlinearinteractionstrengthbetweenthe
+ U −iγ(2) |ψ |2+ i¯hκ(x)n(x) ψ atoms in the trap. The latter can be accomplished by
(cid:16) tu tu (cid:17) u 2 (cid:21) t controlling magnetic field close to a Feshbach resonance
+κ (x)eikxψ , (1) [17]. This is equivalent to controlling the bias magnetic
out u
fieldinamagnetictrap,orapplyingaconstantmagnetic
dψ ¯h2
i¯h u = − ∇2+mgx+ U −i¯hγ(2) |ψ |2 field in an optical trap, and has been achievedwith high
dt (cid:20) 2m (cid:16) uu u (cid:17) u precision [18]. Provided the parameters a1, a2 and b are
+ U −iγ(2) |ψ |2 ψ +κ (x)e−ikxψ ,(2) chosencorrectly,the rate of change of energy of the con-
tu tu t t out t densate must always be non-positive. This conclusion
(cid:16) (cid:17) i
dn assumes that the condensate is not pumped and has no
= r−γ n(x)−κ(x)|ψ |2n(x)+λ∇2n(x), (3)
dt p t losses. Asweshallseelater,thepresenceofsuchfeatures
can mean that this feedback scheme is not always guar-
where m is the atomic mass, V is the trapping poten-
t anteed to remove energy from the condensate, although
tial, g is the acceleration due to gravity (taken to be in
the negative x direction), U = 4π¯h2a /m is the in- in practice it normally still does a very good job.
ij ij
teratomic interaction between ψ and ψ and a is the For our error signals we have chosen the moments hxi
t j ij
s-wave scattering length between those same fields. γ(1) and hx2i, corresponding to the first and second posi-
i tion moments of the condensate, as well as the moment
is the loss rate of ψ due to background gas collisions,
i h|ψ |2i, which we have dubbed “pointiness”.
γ(2) is the loss rate of ψ due to two-body inelastic colli- t
i i In a real system feedback is likely to be limited due
(2)
sions between particles in that state, γ is the loss rate
tu to finite detection speed and the ability to dynamically
of eachfield due to two-body inelastic collisions between
modify the potentials. As with all oscillatory systems
particlesintheotherelectronicstate,κ isthecoupling
out controlled with feedback, when the response time of the
ratebetweenthetrappedfieldandtheoutputbeam,k is
feedback becomes a significant fraction of the smallest
the momentumkick due to the coupling process,r is the
timescaleinthedynamicsofthesystem,thecontrolmay
rate of density increase of the incoherent cloud of atoms
operate as positive feedback. For this reason, it is only
forming the reservoir, γ is the loss rate of that cloud
p safe to use control system where the dynamics of the
and λ is the spatial diffusion coefficient. We choose the
relevant fluctuating moments are within the bandwidth
coupling between the reservoir and the trapped field to
of the feedback. For most BEC systems this is not a
be of the form
problem as a control bandwidth of kiloHertz should be
κ(x)=κ e−x2/σ2 (4) sufficient to respond to fluctuations in the system. The
0
key difficulty in applying feedback in an experimental
enabling us to consider a spatially dependent pumping situationwillbe the destructive effects of the continuous
scheme. detection. Ithasrecentlybeendemonstratedthatoptical
The pumping terms in the above equations are phe- detectioncannotbearbitrarilynon-destructive,andthat
nomenological,describinganirreversiblepumping mech- for a given atomic spontaneous emission rate, increasing
anism from a reservoir which can be depleted but is re- sensitivity over standard techniques requires multi-pass
plenished at a steady rate. These two features are nec- interferometry [19, 20].
essary for any pumping mechanism that generates gain-
We solved equations (1)–(3) both with and without
narrowing through the competition of the gain and loss
feedback terms using a pseudospectral method with a
processesofthelasingmode. Wehavenotincludedthree-
fourth-orderRunge-Kuttatimestep[21]withtheXMDS
body losses, which can be important. Near a Feshbach
numerical package [22] using the atomic properties and
resonancetheymaybenegligible,however,andstillallow
loss rates for 85Rb near a Feshbach resonance, where
a wide range of scattering lengths [16].
the interatomic interaction can be tuned with magnetic
To model the feedback stabilization scheme we adopt
fields. We used the following physically reasonable pa-
ourpreviousapproach[15], andassumethatwecancon-
rameters for all calculations: γ(1) = 7.0 × 10−3s−1,
trol the condensate in some fashion in response to a set t
of continuously measured error signals. Realistic con- γt(2) = 1.7 × 10−8ms−1, γu(1) = 7.0 × 10−3s−1, γu(2) =
trol parameters we can use to affect the condensate will 3.3× 10−9ms−1, γ(2) = 8.3×10−9ms−1, γ = 5.0s−1,
tu p
3
κ = 300, κ = 4.2 × 10−4ms−1, λ = 0.01ms−1,
out 0
k = 1.0× 10−6m−1, with a harmonic trapping poten-
tial of the form V = mω2x2/2 where ω = 50 rad s−1
t
and m = 1.4095×10−25kg. The simulation region was
2.7×10−4m in length. As the interatomic interaction )
strengths between different Zeeman levels are unknown −1m
(
for 85Rb we assume Utt =Uuu =2Utu. sity
n
e
D
III. STABILITY ANALYSIS WITHOUT
FEEDBACK
senFti.rsPtrweveiwouilslacnoanlsyidsiesrhaansashnoawlynsitshwahtetrheefgeeednberaaclkriusleabo-f Position (10 − 5m) Time (s)
stability for pumped atom lasers is that they are unsta-
ble when the interatomic nonlinear interaction strength
is below a certaincritical value, andbecome more stable
as the nonlinear interaction strength increases [11]. Fur-
thermore, the onset of stability occurs earlier (i.e. at a
)
lowernonlinearinteractionstrength)if the condensate is −1m
preferentiallypumpedtowardsthecenterofthetrap;the (
y
more narrowly defined the pumping region in space, the sit
n
more stable is the laser [12]. e
D
Ourmoredetailedanalysisbroadlyconfirmsthesefind-
ings. The general behaviour of the condensate is as fol-
lows: the trap rapidly fills with atoms, reaching a cer-
tainpopulationgovernedby a balancebetweenpumping
atinadllyloflssu.ctTuhaetipnogp,ualnadtiothneasendfluccotnudaetniosantsewdiellnseiittyhearregrionwi- Position (10 − 5m) Time (s)
in magnitude or become damped out over time, depend-
ing on whether the condensate exhibits long-term sta- FIG.1: Generalbehaviourofthecondensateandoutcoupled
bility or not. An example of a borderline stable case is fields. Shown are the condensate density over a two second
displayed in Figure 1, which shows the dynamics of the period (top) and the outcoupled field density over the same
trapped (condensate) and untrapped (outcoupled) field period(bottom). Notethatthefluctuationsoftheuntrapped
densities over a period of two seconds. The condensate field follow those of the trapped field. System parameters
population and density rise rapidly, and then fluctuate were a=1.0×10−10, σ=9.0×10−6m, r=3.7×108s−1.
about a steady state value, and then slowly stabilize.
The density of the outcoupled field is highest near the
outcoupling region, and decreases monotonically in the
negative z direction, reflecting the fact that the atoms and performing a modal analysis. The ground state of
are accelerating under the influence of gravity. The den- the condensate has the minimum energy allowed and is
sity fluctuations of the outcoupled field mimic those of stable in the trap. As the energy of the condensate is
the condensate. increased,itacquirescomponentsofhigherordermodes,
Thedependenceofstabilityontheinteractionstrength for example breathing modes or sloshing modes. Each
is also clear in our simulations, as demonstrated in Fig- mode will experience and gain and loss at a different
ure2. Theplotsshowincreasingstabilityasthenonlinear rate, so between absolute stability and absolute instabil-
interactionstrengthbetweentheatomsisincreased. The ityexistsaregimewheretheenergyofonlyasubsetofall
fact that the dynamics of the untrapped field closely fol- available modes is increasing. This regime is ultimately
lows the fluctuations of the trapped field is again clear. unstable, but can provide a useful distinction.
Consequently we need only examine the stability of the Figure 3 shows an example of the Fourier technique.
condensateinordertodeterminethestabilityoftheatom The quantity of interest is the density of the condensate
beam. at the center of the trap. A time series of this quan-
Determining absolute stability from our simulations tity over a two second simulationis taken, and a Fourier
can be difficult, particularly in borderline cases where decomposition is made of the two periods 1.0s–1.5s and
long timescales are required in order to determine the 1.5s–2.0s. Comparisonofthesetworesultscandetermine
asymptotic behaviour of the system. A good technique whichmodesaregainingenergyandwhicharelosingen-
fordeterminingstabilityistodecomposethedensityfluc- ergy. In the case of Figure 3, the energy in every mode
tuationsofthecondensateintotheirFouriercomponents has decreased in the second time series, indicating that
4
x 10−5 x 10−5 x 108
2 5 3
m) 2.5
n (
sitio 0 0 −1m) 2
Po−20x 10−50.5 1 1.5 2 −50x 10−50.5 1 1.5 2 Density (1.15
2 5
m) 0.5
n ( 0
o 0 0 0 0.5 1 1.5 2
siti Time (s)
o
P 101
−2 −5
0x 10−50.5 1 1.5 2 0x 10−50.5 1 1.5 2
2 5
Position (m)−02 −05 wer (arbitrary units)1100−01
0 0.5 1 1.5 2 0 0.5 1 1.5 2 Po10−2
Time (s) Time (s)
FIG. 2: Increasing the nonlinear interaction leads to stabil- 10−3
10 20 30 40 50 60 70
ity. Left,trappedfielddensity;right,untrappedfielddensity. Frequency (Hz)
Darker regions indicate higher densities. From top to bot-
tom the scattering lengths are a = 0, a = 1.0 ×10−10m, FIG. 3: Central density of the condensate over a two second
a=4.65×10−10m. Otherparameterswereσ=9.0×10−6m, period (top), and the associated frequency power spectrum
r=3.7×108s−1. overtheperiod1.0s–1.5s (bottomleft)and1.5s–2.0s (bottom
right).
for this set of parameters the condensate is stable and is
attracted to the ground state.
Our previous analysis neglected the coupling between
the trappedfield andthe untrappedfieldonthe grounds 10−4 10−4
that the effect of this coupling on the dynamics was
usually negligible. In this approximation the dynamical Unstable
Unstable
emqoudaetsiocnasnhabveeeexvceintedp.ariTtyh,eanodddcopnasreitqyueonftltyheonglryavevitean- σ (m)10−5 σ (m)10−5
tional potential means that odd excitations can be ex-
cited when the coupling is included. The odd modes can Stable Stable
have a higher gain to loss ratio than the even modes, so 10−6 10−6
includingtheoutcouplingcanaffectthedetailsofthebe- 10−12 10−10 10−8 10−12 10−10 10−8
Scattering length (m) Scattering length (m)
haviour of the model. This is particularly evident near
10−4 10−4
the border of stable and unstable behaviour, where the
odd modes can grow over time resulting in instability, Unstable Unstable
whereamodelwithoutoutcouplingwouldpredictstabil-
ity. The precise boundariesdepend sensitively onthe in- σ (m)10−5 σ (m)10−5
teractionof many variables,including the pumping rate.
As we saw in the absence of spatially selective pump- Stable Stable
ing [11],oursimulationsshowthathigherpumping rates 10−6 10−6
lead to greater stability. A stability phase diagram is 10−12 10−10 10−8 10−12 10−10 10−8
Scattering length (m) Scattering length (m)
showninFigure4,showinghowtheboundariesofstabil-
ity and instability are affected by altering the pumping
FIG.4: Theeffectofthepumpingrateonthestabilityofthe
rate. While the long term behaviour becomes asymptot-
laser when feedback is not present. Above the upper line in
ically difficult to integrate near the stability boundaries,
eachplotthelaserisabsolutelyunstable;belowthelowerline
the resolution in our simulations is greater than the size
it is absolutely stable. Between the lines only certain modes
of the features in those boundaries. This means that the arestable. Shownarepumpingratesofr=3.0×107s−1 (top
stability boundaries are not smooth, and are very sensi- left),r=1.0×108s−1m(topright),r=6.0×108s−1 (bottom
tive to the details of the system. left) and r=3.0×109s−1 (bottom right).
Thereisalsotheinterestingpossibilitythatthereexist
stablecondensateconfigurationsthatarenottheground
state. Forexample,Figure5(a)showsanexample where
5
−5 where Vˆ describes the external feedback potential. Hˆ
x 10 fb 0
describes the evolution of the condensate when feedback
2
m) is not present and is given by
n ( 1
sitio 0 Hˆ =Tˆ+V (r)+U |ψ|2, Tˆ=−¯h2 ∇2 (6)
po 0 0 0 2m
p −1
Tra−2 where V0 is the trap potential and U0 the nonlinear in-
teraction strength. Defining the energy of the system as
0 0.5 1 1.5
Time (s) 1
E (ψ)=hTˆ+V i+ U h|ψ|2i (7)
0 0 0
2 2
s)
nit it is possible to show that once feedback is applied, the
u
O. 1.5 rate of change of energy is given by [15]
H.
gy ( 1 ddEt0 =−2im¯h Z Vfb(ψ∗∇2ψ−ψ∇2ψ∗)d3r. (8)
er
n
E0.5 AsmentionedinSectionIwechooseourfeedbackpoten-
0 0.5 1 1.5 2 tial to be given by
Time (s)
V =a (t)x+a (t)x2+b(t)|ψ |2. (9)
fb 1 2 t
FIG.5: Anexampleofasetofparameterswheretheattractor
for the condensate is not the ground state, but rather the If the parameters a (t), a (t), and b(t) are chosen to be
1 2
first excited state with energy 3/2h¯ω per particle. The top
figure shows the density distribution of the condensate over dhxi
a (t) = c , (10)
time, and the lower figure shows the energy per particle in 1 1(cid:20) dt (cid:21)
the condensate. Parameters are a = 0, σ = 10.0×106, and
r=3.7×108. a (t) = c dhx2i , (11)
2 2
(cid:20) dt (cid:21)
dh|ψ |2i
t
b(t) = c (12)
3
(cid:20) dt (cid:21)
the trapped condensate is attracted to the first excited
trapmode. Asthisrepresentsacasewherethescattering
where c , c and c are positive constants and hqi =
1 2 3
length is zero, the energy of the condensate has no non- ∗
ψ qψ dx, then the rate of change of energy becomes
linear component and is simply 3/2h¯ω, the first excited t t
R
state of the harmonic oscillator. dE dhxi 2 dhx2i 2 c dh|ψ |2i 2
0 3 t
=−c −c − .
1 2
dt (cid:20) dt (cid:21) (cid:20) dt (cid:21) 2 (cid:20) dt (cid:21)
IV. STABILITY ANALYSIS WITH FEEDBACK (13)
Thisisclearlynon-positiveanddemonstratesthatinthis
system the energy of the condensate can always be re-
Stabilising the condensate can also be achieved by re-
moved with this feedback scheme.
movingenergyfromitviafeedback. Ifenoughenergycan
In what follows, we will choose the constants to be
be removed the condensate will be in the ground state
and stable, and single-mode operation of the atom laser c = 2 m(mω2+2c dhx2i/dt) (14)
1 2
will automatically follow. We will treat the feedback as
c = mp2ω2/¯h (15)
a set of external potentials which we can continuously 2
modify based on continous measurement of various con- u1 = h¯2/mωN, (16)
densate properties. Ideally we wishto showthat if given
where N is the number of particles in the condensate.
a certain set of controls, it is possible to adjust them in
The values of c and c are chosen to ensure critical
sucha waythat the energyofthe condensatewillalways 1 2
damping of trap oscillations in the absence of any non-
bereduced. AsexplainedinSectionI,itisnecessarythat
linear interaction between atoms, and u is chosen such
we can measure these quantities and modify them on a 1
that it is an experimentally feasible nonlinearity that is
timescale shorter than the shortest relevant timescale of
efficientinremovingenergyfromacondensateinthe ab-
the moments we are using as error signals.
sence of pumping and damping. On the occasions where
We begin by considering the case where the conden-
the term proportional to c in (14) is negative and large
sate is isolated,so that there is no pumping, damping or 2
enoughtocausethequantityunderthesquareroottobe
outcoupling present. In this case condensate evolves via
less than zero, c is set to zero.
2
d The previous analysis becomes more complex, how-
i¯h ψ(r,t)=(Hˆ0+Vˆfb)ψ(r,t) (5) ever,when pumping andloss terms areintroduced. Now
dt
6
whenthedE /dtiscalculatedonefindsthattherearetwo
0 Density without feedback
seperatecontributions,oneinvolvingthefeedbackpoten- 8 1.2
tial and the other involving the pump and loss terms. −1m)
tppTneoeohrntmetedsenseert.mnitatwllCiy,no.ovancoTnosldhvenqeitniurnfigiebrntaushtttlceyioconomntnoshbtteiaranitrbatiewoutnitdooieonoccnfoaounfEispVqbldfueebuadeta;cinoottdonnhssaptithdu(1eemis)rfepetadehnoderdibrnleado(c2esikss)- 8Central density (10 246 Condensate energy 00..168 Energy with feedback
is given by 0 0.4
0 0.5 1 1.5 2 0 0.5 1 1.5 2
dE i Density with feedback
0 ∗ ∗ ∗
dt |fb = ¯hZ (cid:16)Vfb(κoutψtψu−κoutψtψu) −1m) 8 1.2
+ 2im¯h Vfb(ψtddx22ψt∗−ψt∗ddx22ψt)(cid:19)d3r.(17) 8sity (10 46 e energy 0.18
Tbehealswecaoynsdnpoanr-tpoosfi(t1iv7e)iinseoquuralscthoe(m8)e,.wThhicehfiisrsktnpoawrnttoof Central den 2 Condensat 0.6
(17) represents the interaction between the outcoupling 0 0.4
0 0.5 1 1.5 2 0 0.5 1 1.5 2
andthefeedback,andcanbepositiveornegativedepend-
Time (s)
ing on the wavefunctions of the trapped and untrapped
fields.
FIG. 6: The effect of feedback. Top row: central density of
The second contribution to dE /dt is independent of
0 the condensate and its energy when no feedback is present.
the feedback potential and can be broken into six parts;
Bottomrow: centraldensityofthecondensateanditsenergy
five parts proportional to the loss terms γt(1), γu(1), γt(2), when feedback is turned on at t = 0.3s. Note that ground
γ(2),andγ(2),andonepartproportionaltothe pumping state energy for this system is 0.839¯hω per particle after the
u tu
rate κ(x)n(x). The exact form of the terms is lengthy particlenumberhasreachedsteadystate(calculatedusingan
imaginary time algorithm). Parameters used are r = 3.7×
and will not reproduced here. The crucial point is that
108s−1, σ=9.0×10−6m, a=4.65×10−11m.
some of terms can be positive depending on the form
of the trapped and untrapped fields. Consequently the
feedback cannot be guaranteed to reduce the energy of
thecondensateinallcircumstances,althoughthesmaller 102
the pumping, outcoupling and loss terms, the better it
will do.
Although the feedback scheme does not produce sta-
ble steady state operation in all parameter regimes, it Unstable
is still remarkably efficient at stabilising fluctuations in
tbheeincgoanpdpenliseadtet.o Faicgounrdee6nssahtoewwshaincheixsaimnpalereogfiofenedobfapcak- σ (m)101
rameter space where it is normally highly unstable. The
introduction of feedback will often damp the condensate
fluctuations by up to two orders of magnitude or more,
making it useful where all that is required is an “effec-
Stable
tive” stability requirement. That is, if one requires the
atom laser to be in single mode for a short period of 11000−12 10−11 10−10 10−9 10−8
time, say seconds to minutes, then this feedback scheme Scattering length (m)
can be very useful. If, on the other hand, the required
criterion is absolute stability, then this scheme offers a FIG. 7: How adding feedback improves stability. Dashed
marginal improvement. In general, the introduction of line is the boundary of stability with feedback; solid line the
boundary of stability without feedback. Pumping rate was
feedback does not greatly alter the stability class of a
r=6.0×107s−1.
particular set of parameters. If a condensate is unsta-
ble, such that there exist non-groundstate modes whose
energy increases with time, then applying feedback will
drastically damp the oscillations and slow their rate of
V. CONCLUSION
growth,butgiventime the condensatewillstillshowthe
same level of fluctuations as before. Nevertheless, there
are regions of parameter space which show an absolute We have numerically simulated a pumped atom laser,
improvement in stability when feedback is applied. Fig- taking into account the back action of the outcoupled
ure 7 shows an example of such a case. beam as wellas a variety of loss mechanisms,in orderto
7
determine what factorsactto stabilize the laser. As well citeslowerenergymodes atthe expense ofhigherenergy
as considering the effects of altering the pumping enve- modes. In contrast to previous work, we included the
lope, the pump rate, and the atomic scattering length, effect of the backaction of the outcoupled beam on the
we introduced a feedback scheme which dynamically al- condensate, and demonstrated that odd modes are the
tersthetrapparametersandatomicscatteringlengthsin first to become unstable.
order to remove energy from the condensate and reduce Introducingafeedbackschemehadmixedresults. Itis
fluctuations. certainly highly efficient at damping fluctuations in the
As noted in previous work, the three significant de- condensate, leading to approximately single-mode oper-
terminers of stability are the atomic scattering length, ation. Howeverinmanycasesitdoesnotchangethesta-
the pumping rate and the shape of the pumping enve- bilityclassofasetofparameters. Thatis,ifaparticular
lope. This does not change significantly in the presence combination of pump rate, pump envelope and scatter-
of the feedback scheme. Stability increases with scatter- ing length is known to lead to an unstable condensate,
ing length and pumping rate, and also increases as the wherethefluctuationsgrowwithtime,applyingfeedback
width of the pumping region is decreased. The latter will drastically reduce the rate at which the fluctuations
occurs because a narrow pump region preferentially ex- grow, but the system is still ultimately unstable.
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