Table Of ContentIncoherent quantum feedback control of collective light scattering by Bose-Einstein
condensates
Denis A. Ivanov and Tatiana Yu. Ivanova
St. Petersburg State University,
Ulianovskaya 3, Petrodvorets,
St. Petersburg, 198504, Russia
Igor B. Mekhov
University of Oxford, Department of Physics,
Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK
6 It is well known that in the presence of a ring cavity the light scattering from a uniform atomic
1 ensemble can become unstable resulting in the collective atomic recoil lasing. This is the result of
0
a positive feedback due to the cavity. We propose to add an additional electronic feedback loop
2
based on the photodetection of the scattered light. The advantage is a great flexibility in choosing
n the feedback algorithm, since manipulations with electric signals are very well developed. In this
a paperwe address theapplication of such a feedback to atoms in theBose-Einstein condensed state
J and explore the quantum noise due to the incoherent feedback action. We show that although
0 the feedback based on the photodetection does not change the local stability of the initial uniform
1 distribution with respect to small disturbances, it reduces the region of attraction of the uniform
equilibrium. The feedback-induced nonlinearity enables quantum fluctuations to bring the system
] out of the stability region and cause an exponential growth even if the uniform state is globally
h
stable without the feedback. Using numerical solution of the feedback master equation we show
p
that there is no feedback-induced noise in the quadratures of the excited atomic and light modes.
-
t Thefeedbackloop, however,introducesadditional noiseintothenumberof quantaof thesemodes.
n
Importantly, the feedback opens an opportunity to position the modulated BEC inside a cavity as
a
well as tunethephase of scattered light. This can findapplications in precision measurements and
u
quantumsimulations.
q
[
1 I. INTRODUCTION sitiesresultinscatteringoflightintothemoderunningin
v theoppositedirectionwithrespecttothepumpedmode.
0 As a result of the interference between the pumped and
3 Feedback control is known to be a useful tool to mod-
the scattered modes a standing wave optical potential is
2 ify dynamics of classical systems [1]. In resent years the
generatedthatmakesfurther redistributionofthe atoms
2 applicationoffeedbacktoquantumsystemsattractscon-
favorable. FinallytheatomsareredistributedinaBragg
0 siderable attention. Since the appearance of the elabo-
. gratingthateffectivelyscatterthepumpedlightandam-
1 rated theory due to Wiseman and Milburn [2, 3] there
plify the counterpropagating mode. This is similar to a
0 were many theoretical proposals [4, 5] and experimen-
laser.
6 tal realizations [6, 7] of feedback control in optics and
1 atomicphysics. Amongthemmanyfeedback-basedcool- The CARL effect appears as a result of positive feed-
v: ing methods for atoms, molecules and opto-mechanical back due to the ring cavity. The cavity performs the
i systemshavebeendiscussed[8–10]. Mostofthesediscus- feedbackthatisproportionaltotheintegratedsignalover
X
sionsinvolveanegativefeedbackloopasamechanismto some time intervaldefined by the cavityphotonlifetime.
r compensateforthethermalnoise. Howevertherearecir- One, however, can imagine a more elaborated feedback
a
cumstances where the positive feedback can improve the algorithmsuchasproportional-integral-derivativecon-
performanceofanatomicormolecularsystem. Onepos- trol(PID-control)[1], thatcanlead to a superiorperfor-
sibilityistoenhancethecoolingandtheself-organization mance or even introduce some new features into the sys-
inensemblesofparticlescoupledtoacavityfield. Forre- tem dynamics. The simplest possibility is to use anelec-
view of these systems without feedback see [11, 12]. A tronic circuit that processes the photocurrent produced
semi-classical considerations of the application of feed- by the scattered light. In the limit of cold particles the
backaregiveninRef.[13]forcoolingandinRefs[14,15] bandwidth of typical electronic components should be
for the self-organization[16–18]. wide enough to correctly process the photodetected sig-
nal. The feedback mechanism is based on the extraction
In this paper we consider the application of feedback
andprocessionof a classicalsignal,which introduces the
to a Bose-Einstein condensate (BEC) in a configuration
measurement backaction noise into the dynamics. The
similartoacollectiveatomicrecoillaser(CARL)[19–21].
effect of this noise is one of the main subjects of this
ACARLsetupconsistsofatomsinsidearingcavity. The
paper.
cavity is pumped so that only a single running mode is
excited. The uniform distribution of atoms for certain Also in the absence of the feedback the phase of the
pump becomes unstable. The fluctuation of atomic den- scattered CARL light and the position of the atomic
2
Brag-grating are not defined. In the feedback scheme,
basedonthe photocurrentmeasurementofthe scattered
light, the position of the generated periodic pattern of
the atoms as well as the phase of the scattered light will
be locked to the potential that serves as the feedback
actuator.
Suchpositioningofatomsandcontrolofquantumlight
can assist in quantum simulations, based on ultracold
gasesinsidecavities[22–25],whereevensmallshiftinthe
position leads to very different quantum phases. This
is in line with recent breakthrough experiments, where
an external optical lattice trapping quantum gases was
implemented inside a cavity for the first time [26, 27].
TheseexperimentsopenthefieldofcavityQEDfortruly FIG.1. Principal schemeof thesetup. AtomicBEC isinside
many-bodystronglycorrelatedstatesofquantummatter. the ring cavity. The light scattered by BEC form the probe
beamisdetectedandusedtocontroladditionalpotentialvia
We will explore the ultimate quantum regime where
an electronic circuit.
the number of excitations in BEC and the number of
photons are small, so the quantum theory of operation
of the feedback loop [3] will be eventually applied. Us- following result for the atom-field Hamiltonian
ingquantumtrajectories[28,29]aswellassemi-classical ~2
eaqpupaetairosntshwatetshoelvfeeedfobractkhelodoypnmamakicesstohfetshteabsiylisttyemof.thIet H0 =~∆a†a+Z dxψ†(x)h− 2m∂x2 (1)
system dependent on the value of the fluctuations. The +~U a†a+ η 2+a†ηe2ikx+η∗ae−2ikx ψ(x),
0
quantum solution demonstrates that the fluctuations in (cid:0) | | (cid:1)i
thenumberofBECexcitationsgrowwiththevalueofthe where ψ(x) is the bosonic field operator obeying
feedback strength. This is the result of the amplification ψ(x),ψ†(x′) =δ(x x′), ∆ is the atom-field detuning,
−
of the photon number noise by the feedback loop. (cid:2)U0 is the ato(cid:3)m-field interaction constant, and m is the
mass ofthe atom. Apartfrom the mentionedabovesim-
The paper is organized as follows. The description
plified consideration of the cavity, this is the well-known
of the physical model and the mathematics used for it
result that can be found for example in Ref [31].
are given in Sec. II. The results of semi-classical and
Together with the cavity field the controlled potential
fully quantum treatment are presented and discussed in
is applied to the atoms. The control signal determines
Sec. III. The main conclusions of the paper are given in
the depth of this potential. The signal is derived from
Sec. IV.
the result of the measurement of the number of photons
leaving the scattered cavity mode. In principle, one can
designacontrolschemewherethe potentialdepthwould
be any function of the detected signal. We assume the
II. MODEL linear feedback control as it will be shown to be enough
for our purpose. This is described by the following addi-
Although the method is potentially applicable to all tional term to the system Hamiltonian (1)
polarizable particles including molecules [30], we assume
H =i(t τ)Z. (2)
fb
that the particle ensemble is represented by trapped −
atomscooledbelowtheBose-Einsteincondensationtem- Herei(t τ)isthephotocurrentdetectedoutsidethecav-
−
perature. The atom-atom interactions will be neglected ity. The description of feedback can take into account
since quite dilute atomic ensembles, N 104 atoms, will possible delay of the electronic feedback loop, which is
∼
be considered. The atoms are placed inside a ring op- represented by the time interval τ in Eq. (2). Here we
tical cavity, see Fig. 1, and interact with it scattering assume that the photocurrent i(t) is a classical quantity
photons from the externally pumped running mode into thatrepresentstheoutputoftheclassicalapparatus. The
the counter-propagating one. We restrict the considera- feedback operator Z in our case acts only on the atomic
tion to only two counter-propagatingmodes of the same subspace of the total Hilbert space, it represents the ad-
frequency, whichis assumedto be wellred-detuned from ditionalcontrolledpotentialforthe atoms. Itisassumed
the atomic transition. Moreover, as we are interested in to be the standing optical wave with the well defined
the evolution of the atomic sub-system and its ability phase
to form periodic density patterns, we consider regimes
where the field of the coherently pumped mode η can be Z =U K dxψ†(x)sin2(kx)ψ(x). (3)
0
Z
consideredconstant. Thus only one operator cavity field
a remains for the quantum dynamical problem. Adia- The parameter K is the feedback gain, it is defined in
batic elimination of the excited atomic state gives the such a way that the intensity of the lasers forming the
3
feedback potential is given by I (t)=Ki(t τ). The field into a cosine and sine excitation modes as
fb
−
atom-field coupling parameter U here is assumed to be
0
1 2 2
thesameasinEq.(1)sincethesimilarfrequenciesshould
ψ(x)= ψ + ψ cos(2kx)+ ψ sin(2kx). (5)
be used for the probe and feedback light fields. √L 0 rL c rL s
Notethataninterestinggeneralizationofthedescribed
Here L is the length of the BEC sample, the bosonic
feedback setup exists. For the feedback loopone can use
annihilation operators ψ , ψ , and ψ describe uniform
lightscatteredatanyanglewithrespecttotheBECaxis 0 c s
distribution of atoms, their cosine and sine modes, re-
andthe pumpbeamdirection. Thislightcontainsdiffer-
spectively. We assume that the number of atoms in the
ent information about the distribution of the atoms [32–
uniformBECN islargeenoughsothatitonlynegligibly
34] than the back scattered light. Thus generation of
changesdue to the interactionwith the cavity field. The
moreexoticatomicdensitydistributionsandBECstates
large N also allows to neglect the operator character of
may be possible.
ψ and consider ψ =√N.
0 0
To clarify the sortof feedback we are dealing with, we
Using the 3-mode representation of the atomic field,
distinguish between two types of photo-detectors. The
Eq.(5),onecanrewritetheHamiltonian(1)andthemas-
first type is a very fast device that can reproduce all
ter equation in a simpler form. This form of the master
features of the system evolution and response to a sin-
equation will be used for the direct numerical simula-
gle photon. In this case the photo-current is given as,
tions. Havingthis approximationoneobtainsthe follow-
i(t)=dN(t)/dt, the derivative of the total number N(t)
ing Hamiltonian
of photo-counts, which is a Poisson stochastic process,
having the ensemble average E(dN(t))=0 and the mo- H =4~ω ψ†ψ +ψ†ψ +~U √Nη a+a† ψ†+ψ
0 R c c s s 0 c c
bmyenWtiEse(mdNan(t[3)d]Nth(att′)i)n=ohrad†earitδo(tco−rrte′)c.tlIythhaansdbleeetnhisshtoywpne +i~U0√(cid:0)Nη a−a† (cid:1)ψs†+ψs . (cid:0) (cid:1)(cid:0) (6(cid:1))
(cid:0) (cid:1)(cid:0) (cid:1)
of photo-detection one can use the master equation that Here we also define the recoil frequency ω =~k2/2m.
R
in our notation can be written as To eliminate additional phase shift of the optical fields
we assume the atom-field detuning to be given by ∆=
d̺ i
dt =−~[H0,̺]+κD ae−iZ ̺, (4) Ure0wNri.ttTenheuascintigonEqo.f(t5h)etfoeefidnbaalclkypgiovteential Z canalsobe
(cid:2) (cid:3)
w(ch†ce̺re+th̺ec†dc)e/c2ay. sTuhpeerpoaprearmateotrerreκadisstahseDc[acv]̺ity=pch̺oc†to−n ei√N2KU0(ψc†+ψc)ψce−i√N2KU0(ψc†+ψc) =ψc−irN2 KU0.
decay rate. This equation governs the evolution of the (7)
unconditioned density matrix of the whole system, BEC Theexpression(7)isvalidforω U ,otherwiseitwill
R 0
and the cavity field. The equation (4) is valid in the contain addition phase multiplier≫s. This condition can
Markovianlimit, thatis neglecting the time delay τ. We easilybefulfilledasweassumequiteweakcouplingofin-
assumethatintheconsideredsystemtheMarkovianlimit dividualatoms to the cavityfield. At the same time this
can be at least potentially satisfied since the feedback conditiondoes not meanthat the evolutionof the atoms
circuit operates with electro-magnetic signals which are is solely determined by ω since the collective coupling
R
much faster than the motion of atoms. U √N still can be comparable or larger than the re-
0
∼
In contrast to the described above, the other type of coil frequency. To take into account the losses of atoms
feedback uses an integrating detector that averages the from the excited cosine/sine modes we add to the final
measurements over some time interval T. If this inter- version of the master equation (4) the dissipation terms
val is small compared with the atomic evolution time γ [ψ ]+γ [ψ ]. The losses are assumed to have equal
c s
D D
scale and simultaneously large compared with the pho- rate γ. Here we introduce the atomic losses phenomeno-
ton cavitydecay ratethen it is naturalto approximately logicallyandavoiddiscussingtheirphysicalmechanisms.
considerthetimeaverageoverT asanensembleaverage. For the detailed analysis of this aspect refer [35].
The photo currentis thengivenby i(t)=κTr ̺(t)a†a .
InsertingthisintoEq.(3)oneendswithanon-l(cid:8)inearma(cid:9)s-
ter equation. It is rather straightforward to check that III. CLASSICAL AND QUANTUM DYNAMICS
the corresponding equations for average values will con-
tainthe feedbackcontributionasthe potentialforatoms Beforedoingquantumtrajectorysimulationswhichare
proportional to the scattered photon number. No feed- limitedinthenumberoftheexcitationquanta,itisworth
back induced higher-order correlations appear. Such a trying a semi-classical analysis. It should allow to test
feedbackeffectagreeswiththeclassicalintuitionandrep- a broader range of parameters and approximately deter-
resents the physical model of the semi-classical approxi- mine different regimes of the feedback operation. To do
mation that we will also address below. so we calculate the equations of motion for the average
We start with the first type of feedback that pre- atomic and light fields using the master equation (4). It
serves quantum correlations. Following the convenient is convenient to write down these equations in terms of
approachdevelopedinRef.[31]wedecomposetheatomic quadratures, defined as x=(a+a†)/2, y=(a a†)/(2i)
−
4
for the cavity field and analogously for Xc, Yc and Xs, 0.2
Y being the quadratures of the cosine and sine atomic
s 0.0
modes, respectively. The resulting evolution equations )
e
read u−0.2
al
κ
x˙ = 2U0√NηXs x, nv−0.4
− − 2 e
κ g−0.6
y˙ = 2U0√NηXc y, ei
− − 2 (−0.8
γ e
X˙c =4ωRYc Xc, R−1.0
− 2
γ N −1.2
Y˙ = 4ω X Y 2U √Nηx+κKU (x2+y2), 0.0 0.1 0.2 0.3 0.4 0.5
c − R c− 2 c− 0 0r 2 η
γ
X˙ =4ω Y X ,
s R s s
− 2
γ FIG. 2. Real parts of the eigenvalues of the semi-classical
Y˙ = 4ω X Y 2U √Nηy. (8)
s − R s− 2 s− 0 evolution, given by Eqs (8), as functions of the pumping η.
The parameters of the system are U0=0.1ωR, N=104, κ=
Here the feedback enters via the last term in the equa-
10ωR,γ=10ωR.
tion for Y which is proportional to the feedback con-
c
stant K. To obtain this last term it has been assumed
that the average of the intracavity photon number can 2.5
be approximated as the product of the average fields,
K=1.06
a†a = a† a . This is true for coherent cavity field
hstateis, buht wihroing for a general quantum state. 2.0 K=1.05
The easily seensteady state solutionofEqs (8) is zero K=0
field in all of the involved modes. This solution is stable 1.5
for small value of the pumping η, but at some threshold n
valueofη itlosesstabilityandfluctuationsgrowtoinfin- 1.0
ity. Theunphysicalinfinity isthe drawbackofthemodel
andisconnectedtothefactthatthereisnodepletionfor
0.5
BEC and the cavity pump. Thus the considered model
correctly describes the growth of the excitations only in
0.0
the beginning of the evolution.
0 5 10 15 20 25 30
The stability conditions for the zero steady state of
t
Eqs (8) can be analytically calculated but appear to be
quite lengthy. It is much easier to check the stability
for some numerical values. In all the simulations below FIG. 3. The numberof photons scattered from the atoms as
we will use the following values U0 = 0.1ωR, N = 104, afunctionoftimefordifferentvaluesofthefeedbackconstant
κ=10ωR, γ =10ωR. These values correspond to quite K. The solid line corresponds toK=1.06, dashedline corre-
weak coupling and moderate cavity finesse with respect spondstoK=1.05, dashed-dottedline corresponds toK=0.
to realistic cavity QED experiment with cold Rubid- Thepump is set below thethreshold, η=0.3. The time scale
ium [36]. The real parts of the eigenvalues of the evolu- is κ−1.
tionmatrixcorrespondingtoEqs(8)areshowninFig.2.
Itisseenthatforsomevalueofthepumpfieldη 0.3the
≈
real part of one of the eigenvalues becomes positive and below the threshold. The dash-dotted line, that corre-
the zero steady state becomes unstable. Note that the sponds to K=0 thus approacheszerodemonstrating the
presence of feedback does not affect the linear stability decayofinitialfluctuations. Thevaluesoftheinitialfluc-
condition of the uniform equilibrium state as it is repre- tuations are set to quadrature variances of the vacuum
sentedviaaquadratictermandthereforevanishesinthe state: x(0)=y(0)=X (0)=Y =X =Y =0.5. For the
c c s s
linear approximationaroundthe initial equilibrium with value of K =1.05 and the used initial fluctuations the
zero reflected field and atomic excitations. systemis unstable and the rapidgrowthof the scattered
The presence of feedback however changes the range photons n appears, see the dashed line in Fig. 3. The
of attraction of the zero-reflection uniform equilibrium rate of this growthis very sensitive to the value of K, as
state, so that for relatively large values of the feedback itisseenfromthecomparisonofthedashline(K=1.05)
constant K even small fluctuations can force the system andthe solid line (K=1.06). Thus, though the feedback
tolooseitsstability. ThiseffectisdemonstratedinFig.3 based on the photodetection does not change the linear
where the number ofphotons n inthe scatteredmode as stabilityofthezerosteadystate,inthepresenceof(quan-
a functionoftime isshownforseveralvalues ofthe feed- tum) fluctuations the feedback-induced nonlinearity can
back constant K. The cavity pump η is assumed to be render the system unstable.
5
with feedback: K=0.1 No feedback K = 0.0 Feedback K = 0.75
5 11..68 11..68
Q11..42 11..42
nc , n001(cid:0)...680 001...680
4 0(cid:1).4 0.4
n 0.2 0.2
s 0.00 2 4 6 8 10 12 0.00 2 4 6 8 10 12
1.2 0.8
, nncs32 , X∆X000001(cid:1)(cid:0)......4608200 2 4 6 8 10 12 −000000000.........4603751210 2 4 6 8 10 12
1.2 0.8
1.0 0.6
Y
∆0.8 0.4
1 , Y00(cid:0)..46 00..02
0(cid:1).2 −0.2
0.0 −0.4
0 2 4 6 8 10 12 0 2 4 6 8 10 12
0 t t
0 5 10 15 20 25 30
t FIG.5. Theaveragevalues(solidlines)andvariances(dashed
lines) of the number of quanta and the quadratures of the
scattered light. Theleft andright columnscorrespond tothe
FIG. 4. The number of atoms in the ”cosine” nc and ”sine” case without and with the feedback, respectively. The time
ns modes of BEC. The pump is above the threshold, η=0.4, scale is κ−1.
the feedback constant is small, K =0.1. The time scale is
κ−1.
the case above the threshold η = 0.4, but without the
feedback, whereas the right column corresponds to some
Thekey(andthemostpromisingforapplications)dif- feedbackK=0.75actingonthesystembelowthethresh-
ference between the system with and without the feed- old η=0.3. The upper row shows the average number
back is the possibility to control the position of the ofthe scatteredphotons n (solidlines)andtheMandel
atomic Bragg grating and the phase of the scattered parameter Q (dashed linhe)i. The second row shows the
light. The solutions of the classical equations (8) con- average X (solid line) and the variance ∆X (dashed
firmtheabilityofthe feedbacklooptofixthe positionof line) ofthheiamplitude quadrature. Finally,the thirdrow
the atomic density grating. The number of atoms in the correspondstothephasequadrature Y anditsvariance
”cosine” and ”sine” modes of BEC with small feedback, ∆Y. The case without the feedbackhdeimonstrates that
K =0.1, are shown in Fig. 4. It is seen that the feed- the light scattered from the BEC is in the thermal state
back discriminates the ”sine” mode with respect to the asithasnowelldefinedphaseandtheMandelparameter
”cosine” one, since the number of atoms in the ”cosine” equals the averagenumber of photons.
mode grows much faster which eventually results in the
The presence of feedback (left column in Fig. 5) re-
atomic distribution compatible with the feedback poten-
sults in the appearance of some additional noise in the
tial Z. As a consequence the phaseof the lightscattered
number of photons, so that the photon number fluctu-
fromsuchanatomicBragg-gratingwillbelockedtosome
ations exceed that of a thermal light. On the contrary,
certain value. In the absence of the feedback there will
the fluctuations of the quadratures are of the same or-
be no predefined position of the atomic density grating,
der as in the case without the feedback. Moreover, the
its position will be random according to initial density
feedback results in the appearance of nonzero values of
fluctuations.
the quadrature averages, which shows the existence of a
Note that this locking effect cannot be achieved by well-defined phase in the scattered light.
simply imposing some fixed external potential. This po-
What is more interesting is the quantum fluctuations
tentialwillleadtotheoscillationsinthenumberofatoms
of the excited atomic modes. The comparison between
in the excited BEC modes without clear discrimination
the averages and fluctuations of atom number and the
ofoneofthem. Similareffectcanbereachedifonecould
quadraturesfor the ”cosine” mode with and without the
controlthelossesoftheexcitedatomicmodesγandmake
feedback is shown in the Fig. 6. It is worth noting that
them different for the ”sine” and ”cosine” modes. This
even without the feedback the fluctuations of the atom
could be possible, but the physical implementation of
numberarelargerthaninthethermalstate,whichisjus-
such an approachis not obvious.
tifiedby the factthatthe MandelparameterQ is larger
c
Let us now address the quantum behavior of the sys- than the average number of atoms n . The ”sine” and
c
temwithandwithoutthefeedback. We solvethemaster ”cosine”modesinteractwiththelighhtiindependentlyand
equation (4) numerically using the method of quantum donotdirectlyinteractwitheachotherasweneglectthe
trajectories [28, 29]. The simulation has been done with atom-atomscattering. Thus there is anuncertainty con-
the Python package QuTiP [37]. nected with the fact that it is not known which, ”sine”
In Fig. 5 the result of the quantum simulation for the or ”cosine”, was involved in a particular act of the ex-
light mode is shown. The left column corresponds to change of quanta with the light field. This uncertainty
6
No feedback K = 0.0 Feedback K = 0.75 No feedback K = 0.0 Feedback K = 0.75
2.5 4.5 2.5 1.4
, nQcc2011(cid:1)(cid:0)....0505 42302311........00055505 , nQss2011(cid:1)(cid:0)....0505 000011......468202
0.0 0.0 0.0 0.0
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
0.9 0.8 0.9 0.7
Xc00..87 00..67 Xs00..87 00..65
, X∆c00000(cid:0).....46352 0000....4352 , X∆s00000(cid:0).....46352 00000.....40312
0(cid:1).1 0.1 0(cid:1).1 −0.1
0.0 0.0 0.0 −0.2
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
0.9 0.7 0.9
∆Yc000...687 00..65 ∆Ys000...687 00..46
, 00(cid:0)..45 00..43 , 00(cid:0)..45 0.2
Yc00..32 0.2 Ys00..32 0.0
0(cid:1).1 0.1 0(cid:1).1 −0.2
0.0 0.0 0.0
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
t t t t
FIG.6. Theaveragevalues(solidlines)andvariances(dashed FIG.7. Theaveragevalues(solidlines)andvariances(dashed
lines)ofthenumberofquantaandthequadraturesoftheex- lines)ofthenumberofquantaandthequadraturesoftheex-
citedatomic”cosine”mode. Theleftandrightcolumnscorre- cited atomic ”sine” mode. The left and right columns corre-
spondtothecasewithoutandwiththefeedback,respectively. spondtothecasewithoutandwiththefeedback,respectively.
The time scale is κ−1. The time scale is κ−1.
explains the additional noise in the number of atoms of responsible for that.
the atomic modes.
Thefeedback,inaddition,inducesthephaselockingof
the ”cosine” mode, as nonzero average values of the X IV. CONCLUSION
c
and Y quadratures appear. Similar to the photon num-
c
ber in Fig 5 the fluctuations of the quadratures are not The application of the positive feedback loop to BEC
affectedbythefeedbackloopandhaveapproximatelythe with the assistance of a ring cavity has been considered.
same level as in the absence of the feedback. The atom The probe light scattered from BEC into a cavity mode
number fluctuations of the ”cosine” mode are increased is detected and the signal is used to control the poten-
by the feedback, which is similar to the mode represent- tial. Thesystemistreatedbothsemi-classicallyandfully
ing the scattered light. quantum mechanically. The linear stability analysis of
It is left to consider the behavior of the atomic ”sine” a semi-classical model shows that the feedback based
mode. The corresponding averages and the fluctuations on the photodetection does not change the stability of
arepresentedin Fig.7. The results correspondingto the the zero-field equilibrium. However, numerical simula-
casewithoutthefeedbackcoincidewiththeresultsforthe tions show that the phase-space region of attraction of
”cosine”modewithintheaccuracyofthesimulations,left the zero-field equilibrium shrinks due to the feedback-
column in Fig. 6. However, the results corresponding to induced nonlinearity. Thus the system may exhibit ex-
thesystemwiththefeedbackarequitedifferentfromthat ponential growth if due to initial fluctuations it starts
forthe”cosine”mode. Thefactthatthefeedbackaction from some point in the phase space that is sufficiently
is performed by the optical potential that is favorable remote from the equilibrium. In addition the feedback
for the ”cosine” mode results in smaller population of induces a predefined atomic density distribution and as
the ”sine” mode as it is seen from the comparisonof the a consequence a fixed phase of the scattered light.
Figs 6 and 7. Thus the resulting density distribution The quantum treatment of the feedback was based on
of the atoms in BEC will eventually comply with the the Markovian approximation expressed by the master
feedback potential which is predefined. equationofRef.[3]. Thenumericalsolutionofthisequa-
It is important to note that the feedback action al- tionhasshownthatinthepresenceoffeedbackthequan-
most does not affect the quadrature fluctuations of all tumfluctuationscanleadtotheexponentialgrowtheven
the modes, even when the nonzero average values of the belowtheclassicalinstabilitythreshold. Thefluctuations
quadratures appear. The quantity that suffers from the of the quadratures of all the modes are not increased by
additional noise due to the feedback is the number of thepresenceofthefeedback,whichisnotthecaseforthe
quanta. This can be expected since the number of pho- fluctuations of the number of quanta. The photon num-
tons is detected and its noise is amplified in the positive ber fluctuations in the scattered light become stronger
feedback loop. Additional increase in the variances of thaninathermalstateduethenoiseamplificationbythe
atom numbers in the BEC modes is due to the fact that feedback. The atom number fluctuations in the ”sine”
the light mode is separately coupled to the ”sine” and and ”cosine” BEC modes are even larger than that of
”cosine” atomic modes. Thus if a photon appears in the the scattered photon number.
light mode it remains uncertain which atomic mode is The quantum simulation results show that the elec-
7
tronic feedback loop based on the photodetection can V. ACKNOWLEDGMENTS
be applied to control the excitation of specific BEC
modes without the introduction of additional noise in
The authors acknowledge Saint-Petersburg State Uni-
their quadratures. Althoughithas notbeen discussedin
versity for a research grant 11.38.640.2013 and EPSRC
the paper, the use of electronic feedback canbe more ef-
Project EP/I004394/1.
ficient and result in faster growthof BEC nonuniformity
than the cavity-based feedback alone. This can happen
since the electronic feedback strength canbe made quite
large in the beginning of the process. Thus an essential
feedback action on the atoms can be obtained even for
quite few scattered photons. These aspects, thought for A. References
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