A tom -laser coherence and its controlvia feedback
L. K . T hom sen and H . M . W isem an
arXiv:quant-ph/0202027v2 18 Jun 2002
C entre for Q uantum D ynam ics, School of Science,
G ri th U niversity, B risbane, Q ueensland 4111, A ustralia
(D ated: N ovem ber 21,2016)
W e present a quantum -m echanicaltreatm ent ofthe coherence properties ofa single-m ode atom
laser. Speci cally, w e focus on the quantum phase noise of the atom ic eld as expressed by the
rst-order coherence function, for w hich w e derive analytical expressions in various regim es. T he
decay of this function is characterized by the coherence tim e, or its reciprocal, the linew idth. A
crucialcontributor to the linew idth is the collisionalinteraction ofthe atom s. W e nd four distinct
regim es for the linew idth w ith increasing interaction strength. T hese range from the standard
laser linew idth,through quadratic and linear regim es,to another constant regim e due to quantum
revivals of the coherence function. T he laser output is only coherent (B ose degenerate) up to the
linear regim e. H ow ever,w e show that application of a quantum nondem olition m easurem ent and
feedback schem e w illincrease,by m any orders ofm agnitude,the range ofinteraction strengths for
w hich it rem ains coherent.
PA C S num bers: 03.75.Fi,42.50.Lc,03.75.B e
I. IN T R O D U C T IO N
T he invention ofthe laser in the late 1950s[1]created
the eld ofquantum optics and continues to lead to an
enorm ous range of scienti c and technological applications.Itisexpected thatthe realization of\atom lasers"
w ill sim ilarly revolutionize the eld of atom optics [2].
A tom optics is the study ofatom s w here their wavelike
nature becom es im portant, suggesting an analogy w ith
photons [3]. A n atom laser is therefore de ned as a device that produces a continuous beam ofintense,highly
directional, and coherent m atter waves [4], in analogy
w ith the lightproduced by an opticallaser[5]. T he ideal
atom laser beam is a single frequency (i.e., m onochrom atic) de B roglie wave w ith well-de ned intensity and
phase.
T he rst experim ental achievem ents of the B oseEinstein condensation (B EC ) of gaseous atom s [6] was
followed im m ediately by several independent ideas for
creating an atom laser [7]. Since then there have been
experim entaladvances in the coherent release of pulses
[8]and quasicontinuous beam s [9]ofatom s from B EC s,
as well as further theoretical proposals [10]. In the
experim ental con gurations to date, the laser m ode is
the ground state of a trapped B EC , w hich is pum ped
by evaporative cooling of uncondensed atom s, and the
out-coupling (separated in tim e from the pum ping) is
achieved by either R am an orradio-frequency (rf)transitions to an untrapped state. A lthough these experim entalaccom plishm ents do not include sim ultaneous pum ping and output coupling,they do represent m ajor steps
towards achieving an operating atom laser.
R ecentexperim ental[11,12,13]and theoretical[4,14,
15, 16, 17, 18, 19] studies have focused on the fundam ental coherence properties of B EC s and atom lasers.
A tom s (unlike photons) interact w ith each other, producing strong nonlinearities that a ect the coherence of
the trapped condensate and thus also the out-coupled
laser eld. For a single-m ode condensate the dom inant
e ect ofatom ic collisions is to turn num ber uctuations
into uctuations in the energy and hence uctuations in
the frequency,thus causing increased phase uncertainty.
C ollisionalinteractionstherefore lead to a signi cantdecrease in the atom -laser coherence tim e, and a corresponding increase in the linew idth,especially in the case
of B EC s form ed by evaporative cooling. H owever, we
have previously show n that a continuous,quantum nondem olition (Q N D )feedback schem e can e ectively cancel
the linew idth broadening due to such collisions [20].
To study the coherence properties of an atom laser
one can either focuson classicalorquantum noise in the
atom ic eld. Sources ofclassicalnoise m ay be technical,
such as uctuationsin the trapping potential, nite tem peratures,and speci c trap geom etries [12],or dynam ical,such asthree-body recom bination [19]. T he study of
these e ects is usually based on m ean- eld laser m odels
described by G ross-Pitaevskii(G P)-type equations [21].
Q uantum noise is an intrinsic part ofthe atom ic system
as a consequence ofthe uncertainty relations and is the
lim iting contribution to coherence. T he study of this
requires a fully quantum -m echanicalapproach,ofw hich
the m ost com m on is based on the quantum opticalm aster equation [22]. T he com plexities of either approach,
w hich individually require approxim ations to facilitate
theoreticalanalysis,indicate that sim ultaneous analysis
would not be easy [23].
In this paper we present a fully quantum m echanical treatm ent of the coherence of a single-m ode atom
laser and its controlvia the Q N D feedback schem e proposed in R ef.[20]. Speci cally,we study the properties
of the rst-order coherence function (i.e. phase coherence),w hich allow sus to derive the coherence tim e (and
hence laser linew idth) as wellas the power spectrum of
the laser output. Section IIsum m arizes a set ofrequirem entsfor the coherence ofan atom laser rstdetailed in
R ef.[4].Section IIIpresentsourm athem aticalm odelfor
2
the atom laser and show s the resulting linew idth for increasing atom icinteraction strength.N um ericalm ethods
are presented in Sec. III B ,w hile the analyticalresults
are discussed in Sec. III C .W hen the collisionalnonlinearity is very strong, the coherence function undergoes
a quantum collapse and revivalsequence (detailed in III
D ).T hisleadsto an interesting regim ein the powerspectrum (detailed in III E),w hich has not been considered
before.Section IV detailsthee ectsoffeedback based on
a physically reasonableQ N D m easurem entofthecondensate num ber, to include allregim es of the nonlinearity.
Section V concludes.
II. R E Q U IR E M E N T S F O R C O H E R E N C E
T he coherence ofan atom -laser beam can be de ned
analogously to that of an optical laser beam [4]. T he
fundam entalassum ption is that the laser output is well
approxim ated by a highly directional classical wave of
xed intensity and phase,w hich is also ideally restricted
to a single transverse m ode.T he outputshould also be a
stationary process,i.e.,its statistics should be independent of tim e. To be coherent, the laser output should
then additionally have (1) a relatively sm all spread of
longitudinalspatialfrequencies(i.e. be m onochrom atic);
(2) a relatively stable intensity (i.e. be approxim ately
second-ordercoherent);and (3) a relatively stable phase
(i.e.havea relatively slow decay of rst-ordercoherence).
T he rst condition, m onochrom aticity, follow s from
the requirem ent that the laser output approxim ates a
classicalwave. It can,therefore,be expressed in term s
ofthe characteristic coherence length ofthe wave,lcoh =
( k)1 ,i.e.thereciprocalofthespatialfrequency spread.
C ondition (I) becom es
= 2 =k;
lcoh
(2.1)
i.e.,that the coherence length be m uch greater than the
m ean atom ic wavelength [4]. In term s of the spectral
w idth or linew idth, ‘
!, the m onochrom aticity requirem ent sim ply becom es ‘
!,w here the m ean frequency ! is de ned by the kinetic energy ofthe atom s,
i.e.,! = hk2=2m .C ondition (I) also guaranteesthatthe
dispersion ofan atom ic beam w illbe negligible over the
coherence length [4].
To explain the second and third conditions we require
a m any-body description ofthe outputbeam ;see R ef.[4]
for an in-depth discussion. B asically,the output eld of
a laser can be represented by the localized eld annihilation operator b(t),w hich approxim ately satis es the
-function com m utation relation
y
0
[b(t);b (t )]=
(t
0
t);
of xed intensity and phase.T hisisthereforerepresented
m athem atically by
b(t)
ei! t ;
(t)
(2.3)
w here isa com plex num berand the trivialtim e dependence em phasizes that the laser output should be stationary. T he atom ic eld is not exactly a classicalwave
because there are uctuations in the eld am plitude due
to classicaland quantum sourcesofnoise.T hesew illneed
to be sm allorsom ehow cancelled to m axim izecoherence.
It can be argued that there are no m ean elds in both
atom and quantum optics [24],and as such the description ofa laserasa coherentstateisa \convenient ction".
D espitethis,m ean- eld theories(e.g.,based on G P equations) are very successfulin describing the properties of
lasers,as is the use ofinitialcoherent states for solving
and visualizing m asterequations(asdone in thispaper).
A vanishing m ean eld m eans that
hb(t)i= 0:
(2.4)
In thecaseofopticallaserstheoutputm ight,in principle,
be in a state w ith well-de ned am plitude and phase.B ut
since we do notknow the absolutephase,an averageover
allpossible phases gives Eq.(2.4). For atom lasers,only
bilinear com binations of the atom eld are observable
and sim ilarly no H am iltonian is linear in the atom eld
(atom s cannot be created out ofnothing). T hus a m ean
eld am plitude is physically im possible and the absolute
phase is unobservable,again giving Eq.(2.4).
Sincethem ean eld ofa laseriszero,wecannotrequire
hb(t)i = . H owever,for approxim ating b(t) by (t),we
can require that the m ean intensity be given by
hI(t)i= by(t)b(t) = j j2;
(2.5)
and also that the uctuations in intensity should be
sm allin som e sense. T his requirem ent is quanti ed using G lauber’s norm alized second-order coherence function (for a stationary system ) [25]:
g(2)( )= h:I(t+
2
)I(t):i=hI(t)i;
(2.6)
w here the h::i denotes norm alordering.
Fora eld thatissecond-ordercoherent,i.e. g(2)( )=
1, there is no correlation between the arrival tim es of
bosonsata detectorand theirdistribution isPoissonian.
Speci cally the probability for detecting a boson in the
interval(t+ ;t+ + dt) given one detected at tim e t
is g(2)( )hI(t)idt. For the intensity uctuations to be
sm allwe therefore require that [4]
jg(2)( )
1j
1;
(2.7)
(2.2)
at a given point in the output. T hus, I(t) = by(t)b(t)
can be interpreted as the approxim ate atom - ux operator. T he fundam entalassum ption ofa laser is that the
output should be wellapproxim ated by a classicalwave
i.e. the laser output should be approxim ately secondorder coherent: condition (II).
A ssum ing that condition (II) is m et, the intensity of
the laser beam w illbe relatively stable and the only signi cantvariation in the output eld w illbe due to phase
3
uctuations. A usefulm easure ofthe phase uctuations
is the stationary rst-ordercoherence function [25]:
G (1)( )= by(t+
)b(t) ;
(2.8)
or its norm alized form :
g(1)( )= G(1)( )= by(t)b(t) :
(2.9)
U nlike the eld b(t) itself, the bilinear com binations
above are m easurable even for an atom eld. G (1)( )
is sim ply the m ean intensity (2.5) w hen = 0 and as
increases,itdecreasestowardszero asthe phase becom es
decorrelated from its initialvalue at t.
So,the phase ofthe eld m ight be unde ned because
it varies in tim e, i.e., the rst-order coherence decays.
A lthough wecannotexpectthelaserto beapproxim ately
rst-ordercoherentfor alltim e [i.e.,g(1)( ) 1],we can
requirethe decay ofg(1)( )to be slow in som esense.T he
characteristic tim e forthis decay issim ply the coherence
tim e,w hich can be de ned as [4]
Z
1 1 (1)
jg ( )jd :
(2.10)
coh =
2 0
T his is the tim e overw hich the phase ofthe eld is relatively constant.
B ut even if the phase is constant, it m ight also be
unde ned due to a large intrinsic quantum uncertainty
given by
n
1=2 [26]. Since typically n
n,
the quantum phase uncertainty w illbe large ifthe m ean
num ber n is sm all. For the phase to be wellde ned we
thereforeneed the eld to havea largeintensity,i.e.,n
1,over the tim e that the phase ofthe eld is constant,
i.e., for tim es T
coh . T he num ber of bosons in the
output eld fora given duration T is n = hI(t)iT .T hus,
for a well-de ned phase we require hI(t)i coh
1. In
term softhe rst-ordercoherence function thistranslates
to [4]
Z
Z
(1)
hI(t)i jg ( )jd = jG (1)( )jd
1;
(2.11)
w hich quanti es the requirem ent that the decay of rstorder coherence be relatively slow : condition (III).
T his condition for coherence is equivalent to the
requirem ent that the output eld be highly B osedegenerate [4], w hich is rarely considered for optical
lasers because it is so easily satis ed. It requires that
the output atom ux, i.e., hI(t)i, be m uch larger than
the linew idth !. Since the atom -laser linew idth is the
reciprocalofthe coherence tim e,i.e.,
! = 1=coh ;
(2.12)
we nd that condition (I) requires ‘
(III) requires
! and condition
‘
‘
hI(t)i:
(2.13)
Forsingle-m odeopticallaserstypically hI(t)i> ! so condition (III) is always satis ed. O n the other hand,the
collisionalinteractions in atom lasers causes signi cant
linew idth broadening and so satisfying (III) is not guaranteed. H owever,in Sec. IV we show that this broadening can be e ectively cancelled by a Q N D m easurem ent
and feedback schem e.
T he linew idth, or spectral w idth, of a eld is usually de ned as the full w idth at half-m axim um height
(FW H M ) ofthe output power spectrum . In general,the
power spectrum is given by the Fourier transform ofthe
rst-ordercoherence function [22]:
Z1
P (!)=
G (1)( )e i! d :
(2.14)
1
R1
T his is de ned so that 1 P (!)d! = hI(t)i and therefore can be interpreted as the steady-state ux per unit
frequency [27].
If the rst-order coherence function has the form
G (1)( ) / exp(
) then the laser output w ill have a
Lorentzian power spectrum . In this case the FW M H is
exactly equalto thelinew idth asde ned in Eq.(2.12),i.e.
the reciprocalofthe coherence tim e. O n the otherhand,
if G (1)( ) / exp( 2 2 ) then the laser has a G aussian
powerspectrum w ith a FW H M thatisnow only approxim ately equalto the linew idth.
In any case,ifthe rst-ordercoherencefunction hasthe
form G (1)( ) = jG(1)( )jexp(i! ), then condition (III)
for coherence can be restated in term s ofthe m axim um
spectralintensity P (!). From Eqs.(2.14)and (2.10)and
the above assum ption (A ppendix A show s that this is a
good approxim ation for the atom laser),we nd
Z1
jG (1)( )jd = 4hI(t)icoh :
(2.15)
P (!)=
1
T hus, regardless of the resultant shape of the output
power spectrum ,condition (III) becom es
P (!)
1:
(2.16)
N ote that the centralfrequency ! w illbe shifted by any
laserdynam icsw hich cause a rotation ofthe m ean phase
ofthe laser eld.
T he rem aining sections ofthis paper present a study
of the quantum phase dynam ics of the atom laser and
thus the coherence properties ofits output. Speci cally
we study the rst-ordercoherence function asa m easure
ofphase uctuations,w hich (in the absence ofintensity
uctuations) w illbe the lim iting factor to the coherence
tim e ofan atom laser.T hisfunction,asindicated in this
section, is also intrinsically related to the laser output
characteristicsoflinew idth and power spectrum .
III. A T O M -L A SE R L IN E W ID T H
A . A tom -laser dynam ics
T he atom -laserm odelconsists ofa source ofatom s irreversibly coupled to a laserm ode,w hich issupported in
4
a trap thatallow san outputbeam to form .A broadband
reservoiracting both asa pum p and a sink isalso coupled
to the source m odes. T he laser m ode and source can be
m odelled by the ground and excited states ofa trapped
boson eld,representing the condensed and uncondensed
atom s,respectively [4]. G ain can be achieved,for exam ple, by evaporative cooling of the uncondensed atom s.
O ut-coupling from the laser m ode can be accom plished,
forexam ple,by coherently driving condensed atom sinto
an untrapped electronic state. T hism odelcan be sim ply
described by a quantum opticalm aster equation for the
laserm ode alone [4,29],obtained by adiabatically elim inating the source m odesand tracing overthe continuum
of output m odes. T he system is thus characterized by
a wavefunction (r) and annihilation operator a for the
condensate m ode.
Far above threshold, the laser m ode has Poissonian
num ber statistics [30, 31]. In the absence of therm al
or other excess noise,its dynam ics are m odelled by the
com pletely positive m aster equation [4,29]
_=
D [ay]A [ay] 1
+
D [a]
L0 ;
(3.1)
w here the superoperators D and A are de ned as usual
for an arbitrary operator r:
D [r]
y
r r
A [r] ; A [r]
y
1
2 fr r;
g:
(3.2)
T hatthe m asterequation isofthe Lindblad form follow s
from the identity
Z1
y
y
y 1
dqD [aye qaa =2]:
(3.3)
D [a ]A [a ] =
w here L is know n as the Liouvillian for the totalsystem
evolution.
A s m aster equations, Eqs. (3.1) and (3.5) are derived using the B orn-M arkov approxim ation.M ore com plicated m athem atical m odels for the atom laser dynam ics m ay include a full m ultim ode description w ith
non-M arkovian pum ping and/or dam ping (see, e.g.,
R efs. [17, 18]). H owever, only the single-m ode m aster
equation description em ployed in this paperallow sa relatively straightforward analysis.A lthough a single-m ode
schem e,e.g.,Eq.(3.5),ignores the m any source m odes
and the continuum of output m odes, it is the sim plest
physically reasonable m odelfor an atom laser,in that it
includes the essentialm echanism s ofgain,loss and selfinteractions.
T here are also physicaljusti cations for using M arkovian theory. R eference [17]states that the M arkov approxim ation is valid for output coupling rates satisfying
1
Tm ,w here Tm is the output m em ory tim e,w hich
typically ranges from 10 2 to 1 m s. T his condition is
thuseasily satis ed fortypicalvaluesof 1 ,w hich range
from 10 2 to 10 1 s [34]. Furtherm ore,as discussed in
R ef.[18],the B orn-M arkov approxim ation is only valid
for either weak output coupling or large atom ic densities. B EC s form ed by evaporative cooling have strong
atom -atom interactions that correspond to large atom ic
densities. W e are therefore justi ed in m aking the B ornM arkov approxim ation in Eq.(3.5) [but not necessarily
in Eq.(3.1)]. In otherwords,we expectthatstrong nonlinear interactions, rather than any non-M arkovian dynam ics,w illdom inate the linew idth.
0
T he rst term of Eq. (3.1) represents linear output
coupling at rate and the second term represents nonlinear (saturated) pum ping far above threshold, w here
1 is the stationary m ean boson num ber. It is the
decreasing di erence between the gain and loss, as the
laser is pum ped above threshold, that gives rise to the
gain-narrowed laser linew idth [22]. T he localized output
m ode operator b ofthe preceding
p section is then related
to the laser m ode via b = +
a,w here represents
vacuum uctuations [32]. N ote also that we have chosen a reference potentialenergy forthe system such that
there is no H am iltonian / aya in the m aster equation.
To include the e ectsofatom -atom interactionsin the
laserm ode we considera sim ple s-wave scattering m odel
for two-body collisions,w hich is valid for low tem peratures and densities [33]. T his is described by the H am iltonian:
Z
2 has
y y
j (r)j4 d3r; (3.4)
H coll = hC a a aa; C =
m
B . N um erical calculation of linew idth
A s stated in the Sec. II,the coherence tim e ofa laser,
roughly the tim e for the phase of the eld to
becom e uncorrelated w ith its initialvalue. A s show n by
Eq.(2.10),it is determ ined by the stationary rst-order
coherence function (2.9),in w hich the output operators
b can bep replaced by the laser m ode operators a, since
b= +
a. For the evolution described by the m aster
equation (3.5),the coherence function becom es
coh , is
g(1)(t)= Tr[ayeL ta
_=
D [ay]A [ay] 1
+
D [a]
y y
iC [a
a aa; ]
L ;
(3.5)
(3.6)
w here ss is the stationary solution to Eq.(3.5)given by
[30,31]:
X
ss
= e
n
p
w here (r) is the condensate wavefunction and a s is the
s-wave scattering length. T he totalm aster equation for
the laser m ode including atom ic interactions is then
y
ss]=Tr[ ssa a];
n
1
jnihnj=
n!
2
Z
2
d jrei ihrei j; (3.7)
0
. T hus, the state of the laser (3.7) can
w here r =
be thought of either as a m ixture of num ber states or
equivalently a m ixture of coherent states (although see
[35]for a discussion ofthis).
It is a very good approxim ation (see A ppendix A ) to
assum e that ifg(1)(t) is notrealthen its com plex nature
5
is sim ply of the form g(1)(t) = jg(1)(t)jei! t w here ! is
the central frequency of the laser output. T hat is, we
assum e g(1)(t) is com plex due to an e ective detuning
L ! = i![aya; ]in the evolution.T histype ofevolution
causes a rotation ofthe m ean phase proportionalto !,
w hereas the decay of jg(1)(t)j indicates phase di usion.
U sing Eq.(3.6),this allow s the integralin Eq.(2.10) to
be evaluated to give
coh
’
Tr[ay(L
i!)1 a
y
ss]=2Tr[ ssa a]:
(3.8)
Equation (3.8) can be evaluated num erically, for exam ple using the M atlab quantum optics toolbox [36].
T he rst guess for ! is found from the approxim ation
! ’ Im fTr[ayLa
ss]g
!0;
(3.9)
w hich is exact for L = L ! . Subsequent corrections
are found by an iterative procedure. Substituting !k in
Eq.(3.8)gives k ,w hich isthen used to update ourguess
for ! via the expression !k+ 1 = !k
Im (1=2 k ). B asically,this schem e ensures that the calculated coh has a
vanishing im aginary com ponent. T his is justi ed in A ppendix B ,w here we also show that,ifEq.(3.8) is valid,
then only one correction to !0 is needed for an accurate
determ ination of !.
C . A nalytical calculation of linew idth
A nalytically,itiseasierto use the factthatEq.(3.6)is
unchanged if ss is replaced by the initialcoherent state
jrei ihrei jforarbitrary (say = 0).W e thereforehave
g(1)(t)= Tr[ay (t)]=r;
(t)= eL tjrihrj:
this phase evolution is independent of the initialphase
uncertainty,we have
g(1)(t)’ he i
w here h (0)i = h (0)i = r and j i is a coherent eigenstate ofthe laser eld,i.e.,aj i= j i. T he state ofthe
eld atany tim e can thusbe described by the p
probability
distribution for ,or equivalently (since = nei’ ) the
intensity,n = j j2 ,and phase,’,distributions.
T he uctuations in intensity are relatively sm allfor a
laser w ith
1,i.e., n(t)
0. T hen,also assum ing
the num berstatisticsare unchanged by the evolution,we
have n(t) n(t)= n(0),w hich gives
p
h n(t)e i’ (t) i
he i’ (t) i
:
g (t)= p
’
h n(0)e i’ (0) i he i’ (0) i
(1)
(3.12)
N ow the phase distribution at tim e t due to the laser
evolution is given by ’(t) = ’(0)+ (t),i.e.,the phase
distribution of the initial coherent state plus the relative phase di erence (t) = arg[ (t)= (0)]. A ssum ing
ei
1
2
(t)
V (t)
;
(3.13)
In the Q -function representation [22],a density operator has a corresponding Q function de ned by
Q ( ; )= h j j i= ;
(3.15)
R
norm alized such that Q ( ; )d2 = 1. T he action
of an operator on thus has the corresponding m irror
action ofa di erentialoperatoracting on Q ( ; ). T his
isthe m ostconvenientrepresentation forourlaserm odel
because ofthe identity (see A ppendix C )
y
y 1
D [a ]A [a ]
X1
!
k= 1
@
@n
k
Q (n;’);
(3.16)
w hich, since the higher order derivatives are negligible,
can be truncated at k = 2. T his representation also allow susto visualizethedynam icsproduced by them aster
equation (3.5) as show n in Fig.1.
T he m aster equation (3.5) thus turns into a FokkerPlanck equation (FPE) for Q (n;’):
@
1X
Q (z;t) =
@t
2
j;k
(3.10)
(3.11)
i
w here the second approxim ation assum es G aussian
statistics. T he coherence tim e,Eq.(2.10),is thus found
by evaluating the integral:
Z
1
1 1
(3.14)
e 2 V (t)dt:
coh
2 0
X
U sing any suitable phase-space ( ; ) representation,
this expression is then equivalent to
g(1)(t)= h (t)i=h (0)i;
(t)
j
@2
[B jk (z)Q (z;t)]
@zj@zk
@
[A j(z)Q (z;t)];
@zj
(3.17)
w here z = (n;’) and the drift vector A and di usion
m atrix B are given by
2 ( + n) 2nC
:
2nC
=2n
(3.18)
To nd equations ofm otion for the m om ents hzji and
hzjzk i the FPE needs to be converted to an O rnsteinU hlenbeck (O U ) equation. For an O U process,the drift
vector is linear in the variables (n;’) and the di usion
m atrix constant. O ur drift vector A is already linear,
but our B m atrix is not constant. T he sim plest option
is thus to replace allthe am plitudes in B w ith their (Q function) m ean value,i.e.,n !
+ 1. T he equations of
m otion for the m om ents are then dhzji=dt = hA ji and
dhzjzk i=dt= hzjA k i+ hzk A ji+ (B jk + B kj)=2,w here A
and B are now O U param eters.
W e nd thatthe num berstatisticsareunchanged from
thatoftheinitialcoherentstate,w hich fortheQ -function
representation are n = + 1 and Vn = 2 + 1. H owever,
A =
( + 1 n)
; B =
(3 2n)C
6
H owever, by inspection, there are two lim its that can
p
we obtain
be solved analytically. If
Z1
2
2 coh
e ( + 1)t=4 dt= 4 = (1 + 2): (3.23)
0
p
, on the other hand, the rst exponential
For
in Eq. (3.22) is dom inant and then expanding e t to
second order we obtain
Z1
p
2 2 2
t =8
(3.24)
2 coh
e
dt= 2 = :
FIG .1: T ypicalcontour plots ofthe Q function for = 15
and C (
=4 ) = (2 ) 1=2 . Solid ring: stationary laser
state pss, givenp by Eq. (3.7). Solid circle: initial coherent
+ 1j;dashed contour: phase di usion due
state j + 1ih
to laser gain and loss at t 0:8= ;dotted contour: state due
to totalevolution including collisions, i.e., Eq.(3.5), also at
t 0:8= .
the phase-related m om ents are altered. U sing
they are (for the relative phase )
(t) ’
2 C t;
8 C2
V (t) ’
e
2
C n (t) ’
2 C
e
1,
(3.19)
t
t
+
t
1+
2
1:
t; (3.20)
(3.21)
A sexpected,there isa m ean phaseshift(3.19)due to the
collisions,w hile the nonzero covariance (C n = hn i
hnih i)given by Eq.(3.21)explicitly show sthe num berphase correlation produced by the collisions. T he phase
variance(3.20)containstwo term s,w herethe second correspondsto standard laserphasedi usion.T he rstterm
thus indicates the increase in phase uctuations due to
collisions.
T he e ect of collisions on the phase uctuations can
be clearly seen in Fig.1. T his gure show s single contourplotsoftheQ -function forthehypotheticalcoherent
p
state j i and snapshots at a later tim e due to the evolution ofthem asterequation (3.5).Ifweignorecollisions
the e ectofthe laserevolution issim ply phase di usion.
B y including collisions we see two e ects. First,there is
a rotation ofthe m ean phase,due to Eq.(3.19),and second there is phase shearing. T his is due to the nonzero
num ber-phase correlation,Eq.(3.21),indicating that if
the inherentnum ber uctuationsproduce n > n the corresponding phase w illbe less than ,and vice versa.
T he initialcoherent state w illapproach the actuallaser
state ss,Eq.(3.7),as t! 1 .
Substituting Eq.(3.20) into the expression for m agnitude ofthe rst order coherence function (3.13) gives
jg(1)(t)j= e
2
(e
t
+ t 1)=4
e
t=4
;
(3.22)
w here we have introduced = 4 C = as a dim ensionless param eter for the atom ic interaction strength. T his
expression does not have a sim ple analytical solution.
0
T he resultant expression for the atom laser linew idth
due to collisions is thus
p
(1 +p 2)=2 for
p :
‘=
(3.25)
for
2 = 2
C learly,for
1,weobtain thestandard laserlinew idth
‘0 = p=2 [29,30,31]. T hese two expressions agree at
8 = , and they are an excellent t to the nu’
m ericalcalculationsofEq.(3.8),exceptatthe boundary
between the regim es. T his is illustrated by the gure in
ourpreviouspaper[20],and also the extended version in
this paper,Fig.2 appearing in Sec.III D .
Equation (3.25) represents the sam e physicaldynam icsasfound in sim ilarstudiesby the authorsofR efs.[15]
and [37]. Zobay and M eystre [15]present a three-m ode
atom -laser m odel, w ith the output m ode adiabatically
elim inated. Ignoring collisions between pum p and laser
m odes,they obtain phase variances [Eqs. (21) and (22)
of R ef. [15]], w hich are sim ilar and identicalto the exponents ofEqs.(3.23) and (3.24) respectively. See also
the linew idth plotted in Fig. 3 of R ef. [15]. G ardiner
and Zoller[37]studied a B ose-Einstein condensate in dynam icalequilibrium w ith therm alatom s.O ur rst-order
coherence function,Eq.(3.22),hasthe sam e structure as
the analogousexpression,Eq.(184),derived in R ef.[37].
T he two regim es ofEq.(3.25) correspond to the characteristictim e constantsofEqs.(187)and (186)ofR ef.[37]
respectively. T he second ofthese expressions,w here the
nonlinearity is dom inant, is fam iliar as the inverse collapse tim e ofan initialcoherent state in the absence of
pum ping or dam ping [38,39].
Since the output power spectrum is the Fourier transform of G (1)(t) [see Eq. (2.14)], the shape of the spectrum isalso determ ined by theform ofV (t).Forthe two
regim esofEq.(3.25),thelaseroutputhasLorentzian and
G aussian power spectra respectively, as was also found
in R ef. [15]. T hese spectra are illustrated by Fig.6 in
Sec.IV B .See Sec.III C for a m ore in-depth discussion
ofthe atom laser power spectrum .
T he standard laser linew idth (in the absence of collisions) is sim ply given by =2 . For the prelim inary
atom laser experim ents of R efs. [8, 9], the interaction
strength C isalwaysfound to satisfy C
= and hence
1 [34]. A tom lasers,therefore,have a linew idth far
above the standard lim it. Furtherm ore,if > 3=2 the
linew idth w illbe larger than the m ean output ux
.
7
In other words, the atom ic collision strength does not
have to becom e very large before the laser output does
notsatisfy condition (III) forcoherence [i.e.,Eq.(2.13)].
It is thus ofgreat interest to nd m ethods for reducing
the linew idth due to atom ic interactions.O ne m ethod is
continuousQ N D m easurem entand feedback asshow n in
Sec.IV .
T hecoherencefunction clearly hasperiodicrevivalsw hen
t= m =C ,m is an integer.
Including theotherlaserdynam ics,i.e.gain and loss,is
notso straightforward.Since these term s,unlike those in
L C ,are not functions ofthe num ber operatorwe cannot
easily utilize the num ber state representation.T he m ain
e ect,however,is sim ply a decaying envelope applied to
the revivals of Eq. (3.28), such that the strength of C
com pared to
w illdeterm ine the num berofsigni cant
revivals in the coherence function. T his w ill be show n
below . T he revivals of the coherence function becom e
signi cant w hen
4 2 or C
. T his regim e
wasdeterm ined by calculating the exactlinew idth based
on num ericalsolutions of Eq.(3.8) and corresponds to
the interaction strength w here the linew idth begins to
approach a m axim um .
To determ ine the value ofthism axim um linew idth,we
extend the work ofM ilburn and H olm es [40],w ho m odel
an anharm onic oscillator coupled to a zero-tem perature
heat bath,via two basic assum ptions for including saturated gain. T he m aster equation m odelled by M ilburn
and H olm es is (using our notation)
D . R evivals of the coherence function
In the preceding section,the atom -laserlinew idth was
calculated foratom -atom interactionsranging from weak
p
). H owever,exact num erical
(
1) to strong (
calculationsbased on Eq.(3.8)indicated thatthere isan
upperbound to the linew idth (occurring for > 2)that
was not included in the previous analysis. It turns out
that,in addition to linew idth broadening,the collisional
interactionsalso lead to quantum revivals[38]ofthe rst
order coherence function. A lthough, note that in this
very strong collisionalregim e,the output atom ic beam
cannotbe considered a laser according to the de nitions
in Sec. II(since ‘ >
for > 3=2).
To study the regim e of revivals it is helpful to start
by ignoring allother laser dynam ics apart from the collisions. In this case, _ =
iC [ayayaa] = LC ,and we
can analytically solvefortheperiodic structureofg(1)(t).
For an initialcoherent state (0)= j ih j,
g(1)(t)= Tr[ay (t)]= ;
(t)= eL C tj ih j:
_M
(1)
gM H (t)= eiC te
(3.26)
since j j2 =
2iC t
e
);
(3.27)
for the laser,and its m agnitude is
jg(1)(t)j= exp[
g(1)(t)
jg(1)(t)j
(1
cos2C t)]:
eiC te
e
t=4
t=4
exp
(3.28)
exp
1+
(1
1+
i =C )
2 =C 2
2 =C 2
W e only expect this expression to be valid in the very
strong interaction regim e (C
). H ere revivals of
jg(1)(t)jaresigni cantand =C
1 (since
1).A lso,
as show n by Eq.(3.28),revivals occur at m tr = m =C ,
so in the strong regim e the envelope of the coherence
iC [(aya)2 ; ]+
D [a] ;
(3.29)
t=2
(1
1+
exp
i =2C )
2 =4C 2
1
e2iC te
t
(3.30)
To add saturated gain to this m odelwe rst assum e
that,farabovethreshold,thecontribution to phasedi usion isequalforboth gain and loss[41]and so we replace
w ith 2 . Second,including gain w illalm ostcancelthe
overallexponentialdecay e t=2 ofthe coherence due to
loss,resulting in the sm allerterm e t=4 (since thisw ill
give the standard laser linew idth =2 ). T hese assum ptions give the follow ing results for g(1)(t) and jg(1)(t)j:
U sing the num ber state representation, i.e. (t
p) =
P n
2
pm n (t)jnihm j and j i = exp( j j
=2)
jni= n!,
we nd for the rst-order coherence function
(1
=
w hich gives a rst-ordercoherence function ofthe form
P
g(1)(t)= exp
H
1
e2iC te 2
1
e2
t
cos2C t
t
(3.31)
;
C
e2
t
sin 2C t:
(3.32)
function is given by (for nite m )
jg(1)(t)jenv = jg(1)(m tr)j’ exp
’ e 2t=t Q ; tQ = 1=
4
:
+2
m tr
(3.33)
H ere the tim e tQ can be interpreted asthe quantum dissipation tim e,i.e. how long the state would lastifitwas
:
8
in a superposition ofcoherentstates.N onunitary e ects,
such as dam ping,cause a decay of the quantum coherence ofthese states at a rate /
[42]. T his is relevant
because the state produced halfway between revivals by
nonlinear interactions such as L C is in fact a superposition ofcoherent states [43].
T he relationship between the quantum dissipation
tim e and the revivaltim e can be used to give an indication ofthe num ber ofsigni cant revivals in the coherence function fora given interaction strength.T hatis,if
tr
tQ then no revivals w illbe seen,but iftr
tQ as
forthe above equation,the num berofsigni cantrevivals
is oforder tQ =tr. R evivals begin to appear at tr ’ tQ ,
w hich is at ’ 4 2 or C ’
as stated earlier.
W e are now in a position to determ ine coherence tim e
and linew idth in the regim e ofrevivals.From Eq.(2.10),
the coherence tim e is sim ply half the area under the
function jg(1)(t)j. In the very strong interaction regim e
(C
),this area w illbe m ade up ofm any individual
peaksw hich decrease in heightdue to the envelope given
by Eq.(3.33). T he rst of these peaks (w hich actually
starts at jg(1)(0)j= 1) w illbe the sam e as the p
coherence
function for no revivals, and its area w illbe 2 =
as given by Eq.(3.24). T he subsequent peaks w illhave
areas tw ice this area m ultiplied by the height ofthe envelope at that tim e. T hus we have for the totalarea
(Z
)
p
1
X1
2 2
1
2t=t Q
e
2 coh ’
(t m rt)dt
2
0
m =0
!
p
1
X
1
2 2
:
(3.34)
e 2m t r =tQ
=
2
m =0
T hisexpression can be evaluated by
tQ and
P using tr
by noting thatfora geom etricseries, m rm = 1=(1 r).
T he analyticalexpression forthe linew idth in the regim e
ofrevivals is then
p
‘m ax ’ 4 2 3=2:
(3.35)
T his is in exact agreem ent w ith the num erical results
obtained from Eq.(3.6),w hich are plotted Fig.2.
In this gure we have plotted the approxim ate analytical expressions for the linew idth in the absence
[Eqs.(3.25)and (3.35)]and presence [Eq.(4.14)]offeedback (see Sec.IV B for details)as a function ofthe nonlinearity for = 60. W e have also included num ericalresults (see Sec. III B for details) as a com parative
test for the analytical work, w hich is valid for
1.
A s can be seen,the agreem ent is very good even for an
occupation num ber ofonly 60 [44],thus con rm ing the
accuracy ofour analyticalexpressions for the linew idth.
W ithout feedback we see four distinct regim es. T here is
the standard laser linew idth for
1,a quadratic dep
,and a linear regim e for
pendence on for 1
p
2
.T he lattertwo correspond to the regim es
ofEq.(3.25).Finally there isa constantregim e given by
2
Eq.(3.35) for
,w hich is due to the collapses and
revivals ofthe coherence function.
4
10
without feedback
∆ω
2
10
with feedback
linewidth = output flux
0
10
−2
10
0
2
4
6
10 χ = 4µC/κ 10
FIG . 2: A tom laser linew idth in units of for = 1 and
= 60, plotted w ith and w ithout feedback using both analytical (lines) and num erical (points) m ethods. T he dotted
line corresponds to ‘ =
, i.e. interaction strengths w ith
corresponding linew idths below thisline satisfy the coherence
condition of B ose degeneracy [condition (III)].For the feedback results,see Section IV B .
10
10
N ote that the approxim ation used in the num erical
calculation of Eq. (3.8), i.e., g(1)(t) = jg(1)(t)jei! t, is
no longer necessarily valid in the regim e of revivals,
as indicated by the m ulti-com plex-exponentialnature of
Eq.(3.28). N evertheless,our num ericalresults are still
correctbecause we have taken the m ean atom num ber
to be an integer.A trevivalsthe approxim ation to g(1)(t)
becom es
jg(1)(m tr)jei! 0 m tr = jg(1)jei2m
;
(3.36)
w here we have only used the rst guess for !,since the
iterative procedure w illbe inaccurate in this regim e (see
A ppendix B ).T his expression clearly equalsjg(1)jforinteger . T hus, the sam e num erical sim ulation can be
used for allvalues ofC as long as is an integer
E . P ow er spectrum
A s stated in Sec. II,condition (III) for coherence requiresthatthe integralofjG (1)(t)jbe m uch greaterthan
unity [Eq.(2.11)].T hiswasreinterpreted asrequiring the
linew idth to be m uch lessthan the output ux (B ose degeneracy),or equivalently requiring the m axim um spectralintensity to be m uch greater than unity P (!) 1.
N ow the linew idth isonly the FW H M ofthe powerspectrum if it is Lorentzian. A s discussed after Eq.(3.25),
this w illonly be the case in the weak-interaction regim e
p
. A s (or C ) is increased the power spectrum
becom es G aussian, and as enters the very strong interaction regim e itw illno longerhave a sim ple structure
at all. A t these strong values ofthe nonlinearity we see
quantum revivals ofg(1)(t).
In term s of norm alized rst-order coherence function
9
0.4
P(ω)
C=κπµ/4
0.3
(2πµ)−1/2
C=κπµ/2
0.2
C=κπµ
C=4κπµ
C=15κπµ
0.1
0
ω
3
10
4
10
FIG .3: O utput pow er spectrum P (!) w here ! has units of
, plotted for = 15 and various values of C in the strong
interaction regim e. T he dotted line corresponds to the value
4 =‘m ax ,i.e.,the m axim um heightofthe pow er spectrum in
the regim e ofrevivals. N ote the log scale for !.
the power spectrum becom es
Z1
P (!)=
g(1)(t)e i! t dt;
(3.37)
1
w here we have recognized that hIi = byb =
aya =
. From this equation it is clear that as long as g(1)(t)
hasa sim plestructure,i.e.,no revivals,then thespectrum
w illhave a sim ple (Lorenztian orG aussian)lineshape for
a given interaction strength C , w ith the intensity and
w idth determ ined by how fast g(1)(t) decays.
T he m axim um spectralintensity wasde ned in Sec.II
by Eq.(2.15),w hich was based on the assum ption that
g(1)(t) = jg(1)(t)jei! t. Since we are in a reference fram e
w ith zero m ean frequency before including collisions,! is
the detuning frequency due to collisionsw hich causesthe
m ean phase shiftofan initialcoherentstate (see Fig.1).
A tthisfrequency,the outputpowerspectrum w ill,therefore,have a m axim um value given by
P (!)= 4
coh :
(3.38)
A s stated above,in the regim e ofrevivals this approxim ation w illonly be accurate forthe rstguessfor !,i.e.,
!0 = 2 C .
T he linew idth in the strong atom ic interaction regim e
(C > p
) was calculated in the preceding section to
be 4 2 3=2, i.e. Eq. (3.35). T hus, from the above
equation,we expect the m axim um spectralintensity to
reach
p
P (!0)= 4 =‘m ax = 1= 2 ;
(3.39)
as the atom ic interaction strength C is increased far
above
.
Figure 3 illustrates both the sim ple lineshape of the
powerspectrum w hen there are no revivalsand the com plication of the spectrum as C is increased. Since all
the plotted spectra have P (!) < 1,the atom -laser outputclearly doesnotsatisfy condition (III)forcoherence,
Eq. (2.16), in the strong interaction regim e. T he rst
p
spectrum in Fig.3 is in the range
< C <
,and
thus although revivals are not seen, the output is still
above the cuto for B ose degeneracy.
T herem aining spectra illustratetheincreasing e ectof
quantum revivals due to increasing interaction strength.
T he centralpeak of the power spectrum , w hich can be
de ned regardless of revivals as show n in Eq. (
p2.15),
clearly approaches the predicted m axim um of 1= 2 ,
i.e.,Eq.(3.39),for C
. For exam ples ofthe spectrum in the weak interaction regim es ofEq.(3.25),see
Sec.IV B .
IV . R E D U C IN G T H E L IN E W ID T H V IA
FEED B A C K
Section III A showed that the atom ic interactions do
not directly cause phase di usion. R ather, they cause
a shearing of the eld in phase space,w ith higher am plitude elds having higher energy and hence rotating
faster. T he resultant linew idth broadening is a know n
e ect for opticallasers w ith a K err ( (3)) m edium [45].
T he shearing ofthe eld ism anifestin the nite value acquired by the covariance C n (t) in Eq.(3.21). It m eans
that inform ation about the condensate num ber is also
inform ation about the condensate phase. H ence,we can
expect that feedback based on atom num ber m easurem entscould enable the phase dynam icsto be controlled,
and the linew idth reduced.
A . Q N D feedback schem e
Q N D atom num ber m easurem ents can be perform ed
on the condensate in situ via the hom odyne detection of
a far-detuned probe eld [46,47,48,49]. T hisdispersive
interaction causesa phase shiftofthe probe proportional
to the num berofatom sin the condensate.W e considera
far-detuned probe laser beam offrequency !p and crosssectionalarea A thatpassesthrough the condensate [50].
Figure 4 show s an experim entalschem atic for our Q N D
m easurem ent and feedback schem e.
Fora singleatom theinteraction w ith theopticalprobe
eld can be approxim ated by the H am iltonian
V = h
2
p
4
= h
h!p 2
pyp
8A I sat
h pyp;
(4.1)
w here p , , have their usual m eaning and I sat =
2 hc = 3 [51]. H ere p is the annihilation operator for
the probe beam ,norm alized so that h!p pyp is the beam
power.
T he e ective interaction H am iltonian for the w hole
condensate can thus be taken to be
H int = h (aya
)pyp;
(4.2)
w here , de ned in Eq. (4.1), is the phase shift of the
probe eld due to a single atom . H ere we have also subtracted the m ean phase shift such that the probe optical
10
X
FIG .4: Experim entalschem atic. A far-detuned probe laser
of am plitude % interacts w ith the condensate (i.e., the laser
cavity) causing a phase shift in the probe proportional to
the condensate num ber. T he \anti-m ean" lens subtracts the
m ean phase shift leaving the probe w ith a phase shift proportional to the num ber uctuations n. T he photocurrent
Ihom (t) from the hom odyne Y detection of this eld is then
used to m odulate an externalB - eld uniform ly applied to the
condensate.
laser is a m easure ofthe atom num ber uctuations only
(see Fig.4).
T he back action on the condensate due to thisinteraction can beevaluated using thetechniquesofSec.IIIB of
R ef.[52]. A ssum ing the inputprobe eld isin a coherent
state ofam plitude % and m ean powerP ,the evolution of
the atom ic system due the m easurem ent is
_ = %2D [e i
(a y a
)
] ’ M D [aya] ;
(4.3)
w here the the m easurem ent strength is given by
M = %2
2
= P
2
(4.4)
=h!p :
y
T he approxim ation in Eq.(4.3) requires (a a
)
1, w hich for Poissonian num ber uctuations is sim ply
p
1.
T he above result represents decoherence of the atom
laser due to photon num ber uctuations in the probe
eld,resulting in increased phase noise.In a recenttheoreticalstudy by D alvitand co-workers[49],itwasshow n
that dispersive m easurem entsofB EC s cause both phase
di usion [as in Eq.(4.3)]and atom losses. N evertheless,
they also show that phase di usion dom inates the decoherence rate for large atom num bers,i.e.,for
1,
and so the depletion contribution can generally be neglected [53]. Equation (4.3) is equivalent to the corresponding phase di usion term in their work,since it can
be show n thatboth M and the phase di usion rate given
2 5
by Eq.(14) of[49]reduce to
I=hcA 2.
T he e ectofthe interaction (4.2)on the outputprobe
eld isto cause a phase shiftproportionalto the num ber
uctuations,aya
. T he output eld operator is given
by [52]
pout = e i
(a y a
)
i% (aya
);
(4.5)
p
w here again the approxim ation requires
1. H om odyne detection of the Y quadrature of the output
probe eld w illthusbe a m easureofthecondensatenum ber uctuations. T he hom odyne photocurrent operator
is given by [52]
p
Y
): (4.6)
Iout
= ipout + ipyout ’ IiYn
2 M (aya
pin ’ pin
Y
4
3
2
1
0
FIG . 5: C ontour plots of the Q function for = 10 and
C = (2 ) 1=2 (ignoring any m ean phase shifts). B lack
p
p
circle: initialcoherent state j ih j;black ring: stationary
laser state ss, Eq.(3.7). T he other contours correspond to
the evolution at tim e t
1:5= due to successive term s in
the m aster equation. D ashed contour: phase di usion due
to L 0 ; dotted contour: including atom ic collisions, i.e., L 0
and C ;dot-dash contour: including Q N D m easurem ent back
action, i.e., L 0 and C and M , and nally the solid contour
corresponds to Eq. (4.9). A ll contours are for the optim al
p
M = C.
feedback regim e F =
In order to controlthe phase dynam ics ofthe condensate,we w ish to use this hom odyne current to m odulate
its energy. T his can be done,for exam ple,by applying
a uniform m agnetic eld orfar-detuned light eld across
the w hole condensate.In the ideallim itofinstantaneous
feedback,we m odelthis by the H am iltonian
p
H fb (t)= hayaF Ihom (t)= M ;
(4.7)
w here F is the feedback strength and Ihom (t)isthe clasY
sicalphotocurrentcorresponding to the operator Iout
.
T he totalevolution ofthe system including feedback is
obtained by applying the M arkovian theory ofR ef.[52].
T he m aster equation becom es
iC [ayayaa; ]+ M D [aya]
F2
D [aya] ;
+ iF [ayayaa; ]+
M
_ = L0
(4.8)
w here we have allowed for a detection e ciency [52]
and dropped term s corresponding to a frequency shift.
T he term sin Eq.(4.8)describe,respectively,the standard laser gain and loss (L 0 ),the collisionalinteractions
(C ), the m easurem ent back action (M ), the feedback
phase alteration (F ), and the noise introduced by the
feedback. A s before we can visualize the e ect of the
m easurem ent and feedback term s on the evolution ofan
arbitrary coherent state. T his is illustrated by the Q function contours in Fig. 5. In this gure we have ignored the m ean phase shift due to collisionsto m ake the
com parison clearer.
To com pletely rem ove thatunwanted nonlinearity,the
obvious choice for the feedback strength is F = C . W e
also want to m inim ize the phase di usion introduced by
both the m easurem ent and feedback. A weak m easurem entw illgive poorinform ation aboutthe atom num ber,
w ith a high noise-to-signalratio,w hich w illincrease the
11
noisedueto feedback.O n theotherhand,ifthem easurem entistoo strong them easurem entback action itselfw ill
dom inate. T his leads us to guess the optim alregim e for
p
M = C,
both m easurem ent and feedback to be F =
w hich sim ply leaves
_=
D [ay]A [ay] 1
2C
D [a] + p D [aya] :
+
(4.9)
Proceeding as before, we can nd the exact e ect of
the generalfeedback schem e on the atom -laserlinew idth.
N either the m easurem ent nor the feedback a ect the
atom num ber statistics. T he change in the phase statistics are re ected by the new Fokker-Planck equation for
Q (n; ). T he altered term s in the drift vector and diffusion m atrix are A 2 = (3 2n)(C
F ),B12 = B 21 =
2n(C
F ),and B22 = =2n + M + F 2= M . A fter linearizing,we again nd the phase-related m om ents:
(t) ’
+
C n (t) ’
2
2 (C
+M +
F)
e
(4.10)
t
+
t
F2
M
t
1
t;
(4.11)
1;
(4.12)
w here again the approxim ations have used
1.
T hese equations clearly show that all the unwanted
phase statistics are cancelled by choosing a feedback
regim e w ith F = C and furtherm ore the m inim um phase
p
variance is w hen M = C = . Speci cally, both the
m ean phase shift and the correlation between num ber
and phase uctuations are rem oved,and the phase variance is sim ply given by
V (t)=
2
2C
+ p
(4.13)
t:
T hisisexactly the phase variance from the m asterequation (4.9).
In thiscase,Eq.(3.14)hasa sim pleanalyticalsolution:
‘=
1
coh
=
2
1+ p
;
(4.14)
w herewehaveagain used the dim ensionlessatom icinteraction strength = 4 C = .N ote thatunlike Eq.(3.25),
this linew idth is valid for all . T he derivation of
Eq.(4.14)aboveisbased on preselecting thefeedback and
p
m easurem ent param eters (i.e.,F =
M = C ). O n the
otherhand,R ef.[20]presented the analyticalsolution for
the linew idth independentofthe choice ofthese param eters and proceeded to nd the m inim um w ith respect to
the feedback strength. T he result [Eq.(24) ofR ef.[20]]
only di ers from Eq.(4.14) by the term ( 1=4 )=2 .
χ ≈ µ3/2
χ ≈ µ2
χ ≈ µ3/2
2
χ≈µ
P(ω)
10
no feedback
no feedback
with feedback
with feedback
5
0
−50
B . L inew idth results
2 (C
F )t;
8 (C
F )2
V (t) ’
e
2
15
0
50
100
ω 150
200
FIG . 6: O utput pow er spectrum P (!) for = 15 vs ! in
units of ,plotted w ith and w ithout feedback for interaction
strengths of = 3=2 and
= 2 . T he dotted spectrum
( = 3=2 w ith feedback) has a m axim um intensity of 31.
From Fig. 2 (in Sec. III C ) and the above result,
Eq.(4.14),it is evident that our Q N D feedback schem e
o ers a linew idth m uch sm aller than that w ithout feedp <
< 4 2
back for m ost values of . In fact for
thep reduction in linew idth due to feedback is a factor
of 8 = . M ost im portantly, the laser output is B ose
degenerate [satis es condition (III) for coherence],up to
2
3=2
w ith feedback, as opposed to
in the
absence of feedback. T hus, the atom laser w ith feedback rem ains coherent for m uch stronger atom ic nonlinearities than w ithout feedback. It is interesting that
this corresponds to the \conditionally coherent" regim e
3=2
< < 2 as discussed in R ef.[34].
Since the nonlinearity C ise ectively cancelled by this
feedback schem e,the outputpowerspectrum ,Eq.(3.37),
w ill never have the com plicated structure as show n in
Sec.IIIC .A lso,since Eq.(4.13)hasa lineardependence
on tim e (rather than higher powers) the rst-order coherence function decays exponentially and as such w ill
produce a Lorentzian output power spectrum [see discussion after Eq.(2.14)].
T his lineshape is illustrated in Fig.6,w hich also plots
thecorresponding spectra thatwould be produced by the
laser(for the sam e values of ) w ithout feedback.T hese
latter spectra have a G aussian lineshape as discussed after Eq.(3.25). A lso,as indicated by Eq.(4.10),the rotation of the m ean phase due to collision is cancelled
by our feedback schem e, i.e., g(1)(t) is no longer com plex and hence ! = 0.T he powerspectrum forthe atom
laserincluding feedback w illthusbecentered around zero
frequency regardless ofthe atom ic interaction strength,
w hich is con rm ed in the gure.
T he spectra in Fig. 6 are plotted at the two cuto
3=2
values for B ose degeneracy,w hich are
for no
2
feedback and
w hen feedback is included. Ifthe
output is coherent, i.e., satis es B ose degeneracy, then
it clearly has a m uch narrower and thus m ore intense
spectrum than ifthe output is not coherent. T his gure
thus illustrates the link between the three de nitions of
linew idth [discussed after Eq.(2.15)]and its relation to
coherence.T hatis,a narrow intense outputspectralline
12
For the typicalvalues stated above,Eq.(4.15) is indeed
sm all( 10 1 ).
A notherpracticalquestion is,how realistic isthe zerotim e-delay assum ption forthe feedback? Itcan be show n
using the techniques ofR ef.[52]that this assum ption is
justi ed providing the feedback delay tim e is m uch less
than 1 ,the lifetim e ofthe trap due to the outputcoupling.Ifrecentexperim ents[8,9]area usefulguide,trap
lifetim es are oforder 10 2 s [34]. Feedback m uch faster
than thisshould notbea problem .In fact,thetim edelay
could be com pletely elim inated by feeding forward rather
than feeding back. Linew idth reduction can be achieved
equally well by controlling the phase of the atom eld
once it has left the trap as by controlling it inside the
trap (but,ofcourse,an integrated,ratherthan instantaneous,current would be used for the control).
m onochrom atic w ith sm all intensity and phase uctuations. W e used the norm alized rst-order coherence
function g(1)(t) = G (1)(t)=hIi [25] as a m easure of the
phase uctuations.A stincreases,jg(1)(t)jdecreasesfrom
unity asthe phase ofthe eld becom esdecorrelated from
its initial value. Its decay
R 1 is characterized by the coherence tim e coh = 21 0 jg(1)(t)jdt, or by its reciprocal,the linew idth ‘. G (1)(t) also determ ines the output
power spectrum ,w here the peak spectralheight is given
by 4hIi=‘ regardlessoflineshape,w hile fora Lorentzian
the FW H M is also equalto ‘.
W e exam ined the linew idth asa function ofthe dim ensionless atom ic interaction strength, = 4 C = ,w here
is the output coupling rate and C is the atom ic selfenergy. T here are four distinct regim es: the standard
laser linew idth for
1,a quadratic dependence on
p
p
2
for 1
, a linear regim e for
,
2
and nally another constant regim e w hen
. T he
second and third regim es have Lorentzian and G aussian
output spectra respectively. T he last regim e is a consequence of quantum revivals of jg(1)(t)j, w hich are a directe ectofstrong atom ic collisions[38].T hisleadsto a
com plicated structure in the powerspectrum w ith m any
peaks contained in a G aussian-like envelope.
A n im portant condition for atom -laser coherence is
thatthe phase uctuationsbe sm allin a particularsense.
T his is equivalent to requiring B ose degeneracy in the
output,i.e.,that the linew idth ‘ be m uch less than the
the output ux hIi =
. From the results presented
here,thism eansthatthe laseroutputisonly coherentfor
interaction strengthssatisfying < 3=2,i.e.,som ew here
in thethird linew idth regim e.T herefore,collisionsw illbe
a problem for atom -laser coherence,especially for B EC s
form ed by evaporative cooling w here collisions are the
dom inant m echanism . O n the other hand,ifthe atom atom interactionsarestrong enough,thelaseroutputw ill
exhibit the interesting feature ofquantum revivals.
W e also show , expanding upon R ef. [20], that this
linew idth broadening can be signi cantly reduced by a
Q N D feedback schem e. B asically, by feeding back the
results of a Q N D m easurem ent of the num ber uctuations to controlthe condensate energy,it is possible to
com pensate for the linew idth caused by the frequency
uctuations. T he very num ber-phase correlation created
by the collisionsisutilized to canceltheire ect.W e have
show n thatthis linew idth reduction allow sthe outputto
rem ain coherent for interaction strengths up to ’ 2
ratherthan 3=2,w hich isan im provem entby a factorof
p
. For the reasonable param eters ofC
10 2 s 1 and
106, this im provem ent in linew idth is of the order
of103 and in principle could increase coherent values of
= 4 C = from
109 to
1012.
V . SU M M A R Y
A cknow ledgm ents
T he coherence of an atom laser can be de ned [4]
analogously to that of an optical laser: it should be
H .M .W is deeply indebted to W . D . Phillips for the
idea of controlling atom -laser phase uctuations using
corresponds to a long phase coherence tim e, and m ore
speci cally, if the FW H M linew idth is m uch less than
the value of the output ux,then the laser output w ill
satisfy the conditions ofcoherence.
C . E xperim ental realizability
W e w illnow brie y exam ine som e issuesofexperim ental realizability. A question of interest is, how easy is
it to obtain a Q N D m easurem ent of su cient strength
to optim ize the feedback? W e have show n in Sec.IV B
p
that the optim um feedback schem e requires N = C = .
From R ef.[34]we know that m ost current experim ents
work in the regim e w here the T hom as-Ferm iapproxim ation can be m ade,allow ing an analyticalexpression to be
found for [Eq.(6.25)in [34]]and C . Typicalvaluesare
103,
106 and C
10 2 s 1 . To determ ine and
87
henceM weusetypical R b im aging param eters[48,49],
w hich include = 780nm , A
10 11 m 2 ,
5M hz,
2
2G H z, and Isat
10W =m . For these values,
3:3 106 ,and thus M
4:2 104
I. To obtain
a m easurem ent strength ofthe order ofC ,we therefore
require a probe laserintensity ofonly
30W /m2,w hich
is quite reasonable.
A related question is,how m uch ofa problem is atom
loss due to spontaneous em ission by atom s excited by
the detuned probe beam ? T he rate ofthis loss (ignoring
reabsorption)is
(excited population). W e would like
the ratio ofthislossrate to the outputlossrate
to be
sm all. In the optim alfeedback regim e and for
,
this ratio is given by
2
p =4
2
=
4 M 2A Isat
h!p
2A Isat
:
h!p
(4.15)
13
atom num ber m easurem ents.
A P P E N D IX A : SIN G L E C O M P L E X
E X P O N E N T IA L A P P R O X IM A T IO N F O R g(1)(t)
For the m ajority of the calculations in this paper we
assum e that the rst order coherence function can be
given by
g(1)(t)= jg(1)(t)jei! t:
(A 1)
T hat is, we assum e that the com plex nature of g(1)(t)
(w hich is due only to collisions as show n in Sec.III D )
is described by a single com plex exponential for m ost
regim es of the collisionalnonlinearity. W hen the collisionsare strong enough to cause revivalsthisapproxim ation necessarily breaks dow n.
To determ ine the relative errorcaused by this approxim ation we continue the analysisofevolution due to collisions presented in Sec III D ,i.e.,for _ = LC . Speci cally,we quantify the relative di erence between
Z
1 1
e (1 cos2C t) dt
(A 2)
exact =
2 0
question now is,how m uch di erentis approx from exact
using !1?
T he dom inant term in the di erence is sim ply found
by substituting !1 into Eq.(A 5),giving
r
1
:
(A 7)
36 3=2C 2
T he relative size ofthis error term is obtained by com paring to exact,w hich using the Taylorexpansion equals
r
Z
1
1 1 2 C 2 t2
:
(A 8)
e
dt=
’
exact
2 0
4 1=2C 2
T his corresponds to a relative error,given by ( approx
1
. W e have also veri ed
exact)= exact,ofthe order of
the size ofthis error num erically for the fullsystem dynam ics by sim ulations ofjg(1)(t)jand g(1)(t)exp( i!1t)
(here !1 isfound in the sam e way asshow n in A ppendix
B ).T hese results also con rm ed that the single com plex
exponentialapproxim ation breaks dow n w hen C
(or
4 2),as expected.
A P P E N D IX B : IT E R A T IV E P R O C E D U R E F O R
T H E C O H E R E N C E T IM E coh
and
approx
=
1
2
Z
1
e
[1 exp(2iC t)] i! t
dt:
(A 3)
0
T his is done by m ultiple Taylor expansions [ rst of
cos2C tand exp(2iC t),and then expanding resultantexponentials apart from exp(at2)]and using the equality
Z1
2 2
[(n + 1)=2]
;
(A 4)
tn e 2 C t dt=
2(
2
C 2)(n + 1)=2
0
w here [n]is the G am m a function.
N ow , approx di ersfrom exact by both realand im aginary term s.T he dom inant[fort< ( C 2 ) 1=2 ]realterm
is
r
3=2
1
(2 C
!)2
+ :::
(A 5)
32
2
C2
A sstated in Sec.IIIB ,the coherence tim e can be evaluated num erically using Eq.(3.8),w here the rst guess
for ! is given by Eq.(3.9). Subsequent corrections to !
are found by applying the procedure:
k
!k+ 1
A t each step we have
Z
2k=
1
+ :::
!)
8( C 2)2
(A 6)
W e can thus determ ine ! by requiring that the im aginary term s vanish. T his is essentially w hat the iterative
procedure ofA ppendix B achieves.
T he series of im aginary term s above leads to a rst
choice of!0 = 2 C ,i.e.,this sets the rst term to zero
and we are left w ith term s ofO [( C ) 1 ]. To cancelthe
rst two im aginary term we require 2 C
! = 2C =3,
leaving term s ofO [( 2 C ) 1 ]. H ence,!1 = 2 C
2C =3
w illlargely ensure that approx isreal(since
1).T he
1
2 3
C +
(2 C
3
12
3
ss]=2
aya ;
(B 1)
1
g(1)(t)e i!
kt
dt;
(B 2)
0
w hich is sim ply a reexpression ofEq.(B 1). T hen using
theassum ption ofA ppendix A thatg(1)(t) jg(1)(t)jei! t,
we have
Z1
2k
jg(1)(t)jei(! ! k )tdt:
(B 3)
0
w hile the rst three im aginary term s are
(2 C
!)
8 C2
=
Tr[ay(L
i!k ) 1 a
= !k
Im (1=2 k ):
T he sim plest form for jg(1)(t)jis a decaying exponential, exp( t), w hich gives rise to a Lorentzian power
spectrum as discussed at the end ofSec.II.In this case
Z1
1
2k
e [ i(! ! k )]tdt=
; (B 4)
i
(
! !
k)
0
and hence
Im (1=2 k )
!k
!:
(B 5)
T he m ore com plicated form of jg(1)(t)j = exp( 2t2),
w hich givesriseto a G aussian powerspectrum ,also obeys
14
Eq.(B 5),since in this case
Z1
2 2
2k
e [ t i(! ! k )t]dt
p0
2
2
e (! ! k ) =4 erfc
=
2
’
1=i(! !k ):
C om bining these leads to the follow ing correspondence
for the saturated gain term :
i(!
2
D [ay]A [ay] 1
!k )
(B 6)
T hus,ifthe coherence function is in the regim e w here
the output spectrum is Lorentzian or G aussian, then
!k+ 1
!,and the rst correction,!1,w illbe su cient
for an accurate calculation. O n the other hand, if the
coherence function is in the regim e ofrevivals then the
approxim ation g(1)(t)
jg(1)(t)jei! t itself is no longer
valid. T herefore, the above num erical m ethod w ill be
w ildly inaccurate and we require anotherm ethod forobtaining thecoherencetim eand linew idth.T hisisdetailed
in Sec.IIID .
D [ay] !
A [ay] !
@
n Q (n;’);
@n
@
n+
n Q (n;’):
@n
(C 1)
1
@
@
n n+
n
@n
@n
Q (n;’); (C 3)
w hich can be expanded in two di erent ways to give
(
1
1 @
n
1+
n @n
)
1 Q (n;’);
(C 4)
or
@
@n
1+
1 @
n
n @n
1
Q (n;’):
(C 5)
Equating these expressions leads to
A P P E N D IX C : Q -F U N C T IO N
C O R R E SP O N D E N C E F O R SA T U R A T E D G A IN
A s stated in Sec. IIIC the m asterequation can be reexpressed as a Fokker-Planck equation for a convenient
probability distribution. In this appendix we show that,
for the Q function,the operator correspondence for the
gain term is given by Eq.(3.16). T he individualsuperoperators in this expression have the corresponding differentialoperators,
!
1+
1 @
n
n @n
1
=
1+
@
@n
1
;
(C 6)
and hence
(
)
1
@
D [ay]A [ay] 1 !
1+
1 Q (n;’): (C 7)
@n
B y then applying the Taylor expansion, (1 + x) 1 =
P
1
k
k= 0 ( x) ,we obtain the identity
D [ay]A [ay] 1
X1
!
k= 1
@
@n
k
Q (n;’):
(C 8)
(C 2)
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[50] For sim plicity,w e w illassum e that the distortion ofthe
beam front,and the m ean phase shift,are rem oved by a
suitable \anti-m ean-condensate" lens.
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[53] T he depletion rate scales as M ,so this w illnot be negligible com pared to the output loss rate w hen M > .
For the optim alregim e, this w ould occur w hen C > ,
that is > .A s argued in [34],this is not the expected
regim e of operation for an atom laser. N evertheless, it
m eans that the higher regions of the feedback curve in
Fig.2 should be taken w ith a grain ofsalt.