Effect of one-, two-, and three-body atom loss processes on superpositions of phase states in Bose-Josephson junctions

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by  Dominique Spehner

Effect of one-, two-, and three-body atom loss processes on superpositions of phase
states in Bose-Josephson junctions
D. Spehner,1,2,3,∗ K. Pawlowski,4,5,6 G. Ferrini,7 and A. Minguzzi2,3
1Univ. Grenoble Alpes and CNRS, Institut Fourier, F-38000 Grenoble, France
2Univ. Grenoble Alpes, LPMMC, F-38000 Grenoble, France
3CNRS, LPMMC, F-38000 Grenoble, France
4Center for Theoretical Physics PAN, 02-668 Warsaw, Poland
4 55.Physikalisches Institut, Universität Stuttgart, D-70569 Stuttgart, Germany
1 6Laboratoire Kastler Brossel, Ecole Normale Supérieure, F-75231 Paris, France
0 7Laboratoire Kastler Brossel, Université Pierre et Marie Curie, F-75000 Paris, France
2 (Dated: June27, 2014)

n
In a two-mode Bose-Josephson junction formed by a binary mixture of ultracold atoms, macro-
u
scopicsuperpositionsof phasestatesareproducedduringthetimeevolutionafter asuddenquench
J
to zero of the coupling amplitude. Using quantum trajectories and an exact diagonalization of the

6 master equation, we study the effect of one-, two-, and three-body atom losses on the superposi-
2 tionsbyanalyzingseparatelytheamountofquantumcorrelationsineachsubspacewithfixedatom

number. Thequantumcorrelationsusefulforatominterferometryareestimatedusingthequantum
]
s Fisher information. Weidentify thechoice of parameters leading tothe largest Fisher information,
a therebyshowingthat,forallkindsoflossprocesses,quantumcorrelationscanbepartiallyprotected
g from decoherence when the losses are strongly asymmetric in thetwo modes.
-
t
n PACSnumbers: 03.75Gg, 42.50.Lc,03.75.Mn,67.85.Hj
a
u
q I. INTRODUCTION multi-component superpositions of CSs [9, 10], and then
. has a revival in the initial CS.
t
a
m Non-classicalstatessuchassqueezedstatesandmacro- To date, only squeezed states, which appear at times
scopic superpositions of coherent states are particularly much shorter than the revival time, have been realized
-
d interesting for high-precision interferometry since they experimentally [11–13]. At longer times, recombination
n allowfor phase resolutionbeyondthe standardquantum and collision processes leading to losses of atoms in the
o limit. One of the systems where such states may be en- BEC give rise to strong decoherence effects and eventu-
c gineered is a Bose-Einstein condensate (BEC) made of allytothedisappearanceoftheBEC.Particlelossesalso
[
metastable vapors of ultracold atoms. This system dis- destroy partially the coherence of the squeezed states,

2 plays a wide tunability of parameters: the interaction as analyzed quantitatively in [14–17]. The phase noise
v betweenatomscanbecontrolledbyFeshbachresonances produced by magnetic fluctuations in internal BJJs is
8 [1, 2], and, by using optical lattices, the BEC can be another important source of decoherence [12, 18]. The
3
coherently split into up to few thousands sub-systems superpositions of CSs appear later in the evolution and
2
withcontrolledtunnelingbetweenthem[3–5]. Whenthe areexpectedtobemorefragilethansqueezedstates. The
7
condensed atoms are trapped in a double-well potential, main theoretical studies on decoherence effects on such
.
1
they realize an external Bose-Josephson junction (BJJ). superpositions have focused on the influence of the cou-
0
The spatial wave functions localized inside a single well pling of the atoms with the electromagnetic vacuum [19]
4
constitute the two modes of the BJJ and the tunneling and the impact of phase noise [20]. In particular, it has
1
: between the wells leads to an inter-mode coupling. An beenshownin[18,20]thatthecoherencesofthesuperpo-
v
internal BJJ is formed by condensed atoms in two hy- sitionsarenotstronglydegradedbyphasenoise,andthis
i
X perfine states resonantly coupled by a microwave radio- degradationdoes not increase with the number of atoms
frequency field, trapped in a single harmonic well. In in the BJJ.Under currentexperimental conditions, pho-
r
a bothcases,wheninter-modecouplingdominatesinterac- ton scattering is typically negligible and phase noise can
tions,thegroundstateoftheBJJisaspincoherentstate bedecreasedbyusingaspin-echotechnique[13]. Insuch
(CS), that is, a product state in which all atoms are in conditions, the most important source of decoherence is
thesamesuperpositionofthetwomodes. Afterasudden particle losses. Three kinds of loss processes may play a
quench to zero of the coupling, the dynamical evolution role: one-body losses, due to inelastic collisions between
buildsupentangledstatesbecauseoftheinteractionsbe- trappedatomsandthebackgroundgas;two-bodylosses,
tweenatoms. Intheabsenceofdecoherencemechanisms, resulting from scattering of two atoms in the magnetic
thesystemevolvesfirstintosqueezedstates[6–8],thento trap, which changes their spin and gives them enough
kinetic energy to be ejected from the trap; and three-
body losses, where a three-body collisionevent produces
a molecule and ejects a third atom out of the trap.
∗ [email protected] Inapreviouswork[21],wehaveanalyzedtheimpactof
2
two-body losses on the superpositions of coherent states II. MODEL AND METHODS
produced in internal BJJs. In this paper, we extend this
analysisandstudy the combinedeffectofone-body,two- A. Quenched dynamics of a Bose-Josephson
body, and three-body losses on the formation of the su- Junction
perposition states. By using a quantum trajectory ap-
proach we find, in agreement with Ref. [22], that for all
In this subsection we first recall the main features of
types of losses the fluctuations in the atomic interaction
the dynamics of a two-mode Bose-Josephson junction
energy produced by the random loss events give rise to
(BJJ) in the quantum regime after a sudden quench of
an effective phase noise. We show that for weak loss
the inter-mode coupling to zero. We then introduce the
rates this noise is responsible for the strongest decoher-
Markovianmasterequationdescribingatomlossesinthe
ence effect. The tunability of the scattering lengths by
BJJandtheconditionaldensitymatriceswithfixednum-
Feshbach resonances makes it possible to switch the ef-
bers of atoms.
fective phase noise off in the mode loosing more atoms,
without changing the interaction strength in the unitary
dynamics, i.e.keeping the formation times of the super- 1. Initial coherent state and Husimi distribution
positionsfixed. Onemayinthiswaypartiallyprotectthe
coherencesofthesuperpositionsforstronglyasymmetric Wedenotebyaˆ ,aˆ†,andnˆ =aˆ†aˆ thebosonicannihi-
losses in the two modes, as it has been already pointed i i i i i
lation, creation, and number operators in mode i=1,2.
outinRef.[21]inthecaseoftwo-bodylosses. Weshowin
The total number of atoms in the BJJ is given by the
thisworkthatthisresultappliestoalllossprocessesand operator Nˆ = nˆ +nˆ . The Fock states n ,n are the
that for moderate lossrates the correspondingstates are 1 2 | 1 2i
joint eigenstates of nˆ and nˆ with eigenvalues n and
1 2 1
more useful for high-precision atom interferometry than
n , respectively. Initially, the BJJ is in its ground state
the squeezedstates. This usefulnessforinterferometryis 2
in the regime where inter-mode coupling dominates in-
quantified by the quantum Fisher information F, which
teractions. This initial state is well approximated by
is related to the best phase precision achievable in one
measurement according to (∆ϕ) = 1/√F [23]. We π
best ψ(0) = N ;φ=0 N ;θ = ,φ=0 , (1)
calculate the Fisher information as a function of time in | i | 0 i≡| 0 2 i
the lossy BJJ by using an exact diagonalization of the
where N is the initial number of atoms and
master equation. 0
N N 1/2 (tan(θ/2))n1
N;θ,φ =
| i nX1=0(cid:18)n1 (cid:19) [1+tan2(θ/2)]N2
e−in1φ n ,n =N n (2)
1 2 1
| − i
are the SU(2)-coherent states (CSs) for N atoms [24].
Anarbitrary(pureormixed)stateρˆwithN atomscan
be represented by its Husimi distribution on the Bloch
The paper is organized as follows. In Sec. II we re-
sphere of radius N/2,
call the dynamical evolution in a BJJ in the absence of
tunneling and present the theoretical tools used to ana- 1
Q (θ,φ)= N;θ,φρˆN;θ,φ .
lyze it. We first introduce the Bose-Hubbard model and N πh | | i
the Markovianmaster equation describing the dynamics
This distribution provides a useful information on the
in the presence of particle losses (Sec. IIA). In the re-
phase content of ρˆ. The initial CS (1) has a Husimi dis-
maining part of the section, we give a brief account on
tribution with a single peak at (θ,φ) = (π,0) of width
atominterferometry(Sec.IIB) andonthe quantumtra- 2
jectory method for solving master equations (Sec. IIC). 1/√N0, as shown in the panel (a) of Fig. 1.
≈
Our main results on the time evolution of the quan-
tum Fisher information in a lossy BJJ are presented in
2. Dynamics in the absence of atom losses
Sec. III. These results are explained in Sec. IV with the
help of the quantum trajectory approach. We analyze
separately the contributions to the total atomic density After a sudden quench to zero of the inter-mode cou-
matrix of quantum trajectories which do not experience pling at time t=0, the two-mode Bose-HubbardHamil-
any loss (Sec. IVB) and of trajectories having a single tonian of the atoms reads [25]
or several loss events (Sec. IVC). The various physical
U
effects leading to an increase or a decrease of the Fisher Hˆ = E nˆ + inˆ (nˆ 1) +U nˆ nˆ , (3)
0 i i i i 12 1 2
information at the formation times of the macroscopic 2 −
i=1,2(cid:18) (cid:19)
superpositions are described in detail (Sec. IVD). Sec- X
tionVcontainsasummaryandconclusiveremarks. Four where E is the energy of the mode i, U the interac-
i i
appendices offer some additional technical details. tion energy between atoms in the same mode i, and U
12
3
the interaction energy between atoms in different modes (a) (b)
π π
(U =0 for external BJJs). For a fixed total number of
12
atomsN =nˆ +nˆ ,theHamiltonian(3)hasaquadratic
0 1 2
terminthe relativenumberoperatornˆ1−nˆ2 ofthe form θ θ
χ(nˆ nˆ )2/4, with the effective interaction energy
1 2
−
χ= U1+U2−2U12 . (4) 0-π π 0-π π
2 φ φ
The atomic state ψ(0)(t) = e−itHˆ0 ψ(0) displays a pe- (c) (d)
| i | i π π
riodic evolution with period T = 2π/χ if N is even
0
and T/2 if N is odd. Before the revival, the dynam-
0
ics drives the system first into squeezed states at times θ θ
t TN−23 [6](seepanel(b)inFig.1). Atthelatertimes
≈ 0
0 0
π T -π π -π π
t = = , q =2,3,..., (5) φ φ
q
χq 2q
the atoms are in macroscopic superpositions of coherent FIG. 1. (Color online) Husimi functions in the absence of
states, losses in the BJJ at some specific times: (a) t= 0 (coherent
state); (b) t = T/40 (spin squeezed state) , (c) t = T/6
q−1 (3-component superposition of phase states), (d)t=T/4 (2-
|ψ(0)(tq)i= ck,q N0;φk,q , (6) componentsuperposition). Otherparameters: U1 =U2 = 2Tπ,
Xk=0 (cid:12) (cid:11) U12 =0, E1=E2 =0, and N0 =10.
(cid:12)
with coefficients c of equal moduli q−1/2 and phases
k,q
θ =π/2 and φ =φ +2πk/q, where φ depends on
k,q 0,q 0,q
q,N ,andtheenergiesE andU [9,10]. Inparticular,at
0 i i Thesediagonalandoff-diagonalpartsexhibitremarkable
time t =t the BJJ is in the superposition (N ;φ
2 | 0 0,2i− structures in the Fock basis, which allow to read them
N0;φ1,2 )/√2 of two CSs located on the equator of the easily from the total density matrix [18]:
| i
Blochsphereatdiametricallyoppositepoints. Panels(c)
and (d) of Fig. 1 show the Husimi distributions of the
states (6) for q =2 and q =3. n ,n [ρˆ(0)(t )] n′,n′ = 0 if n′ =n modulo q
Itiseasyto determinethe matrixelements ofthe den- h 1 2| q d| 1 2i 1 6 1
sity matrix ρˆ(0)(t) = ψ(0)(t) ψ(0)(t) in the Fock basis. n ,n [ρˆ(0)(t )] n′,n′ = 0 if n′ =n modulo q.
They have time-indep|endentimh oduli | h 1 2| q od| 1 2i 1 1 (10)
The off-diagonal part does almost not contribute to the
n ,n ρˆ(0)(t)n′,n′ = 1 N0 1/2 N0 1/2 Husimidistribution. TheHusimiplotsinFig.1(c,d)thus
|h 1 2| | 1 2i| 2N0 (cid:18) n1 (cid:19) (cid:18) n′1 (cid:19) essentially show the diagonal parts only. On the other
(7) hand,the quantumcorrelationsusefulfor interferometry
behaving in the limit N0 1 like (i.e.giving rise to high values of the Fisher information,
≫
see below) are contained in the off-diagonalpart [18].
2 exp 1 n N0 2+ n′ N0 2 , (8)
πN −N 1− 2 1− 2
r 0 0
n (cid:16)(cid:0) (cid:1) (cid:0) (cid:1) (cid:17)o
where we have set n =N n and n′ =N n′.
2 0− 1 2 0− 1
Atthe time t offormationofthe superposition(6), it
q
is convenientto decompose ρˆ(0)(tq)as a sum ofa “diago- 3. Master equation in the presence of atom losses
nalpart” [ρˆ(0)(t )] ,correspondingtothestatisticalmix-
q d
tureoftheCSsinthesuperposition,andan“off-diagonal
part” [ρˆ(0)(t )] describingthecoherencesbetweenthese WeaccountforlossprocessesintheBJJbyconsidering
q od
CSs. Defining[ρˆ(0)(t )] =c c∗ N ;φ N ;φ , the Markovianmaster equation [26–28]
q kk′ k,q k′,q| 0 k,qih 0 k′,q|
one has [18]
q−1 dρˆ
[ρˆ(0)(tq)]d = [ρˆ(0)(tq)]kk dt =−i Hˆ0,ρˆ(t) +(L1-body+L2-body+L3-body)(ρˆ(t))
Xk=q0−1 where we(cid:2)have set(cid:3)~ = 1, ρˆ(t) is the atomic density(m11a)-
[ρˆ(0)(tq)]od = [ρˆ(0)(tq)]kk′ . (9) trix,andthesuperoperatorsL1-body,L2-body,andL3-body
describe one-body, two-body, and three-body losses, re-
k6=k′=0
X
4
spectively. They are given by measurementofNˆ is still an experimentalchallenge,the
precisionhasincreasedbyordersofmagnitudeduringthe
1
(ρˆ)= α aˆ ρˆaˆ† nˆ ,ρˆ last years [29–31].
L1-body i i i − 2 i
iX=1,2 (cid:16) (cid:8) (cid:9)(cid:17)
1
L2-body(ρˆ)= γij aˆiaˆjρˆaˆ†iaˆ†j − 2 aˆ†iaˆ†jaˆiaˆj,ρˆ B. Quantum correlations useful for interferometry
1≤Xi≤j≤2 (cid:16) (cid:8) (cid:9)(cid:17)
(ρˆ)= κ aˆ aˆ aˆ ρˆaˆ†aˆ†aˆ† A useful quantity characterizingquantum correlations
L3-body ijk i j k i j k (QCs)betweenparticlesinsystemsinvolvingmanyatoms
1≤i≤Xj≤k≤2 (cid:16) is the quantum Fisher information. Let us recall briefly
1
aˆ†aˆ†aˆ†aˆ aˆ aˆ ,ρˆ , (12) its definition and its link with phase estimation in atom
−2 i j k i j k
interferometry (see [18, 32, 33] for more detail). In a
(cid:8) (cid:9)(cid:17)
where the rates αi, γij, and κijk correspond to the loss Mach-Zehnder atom interferometer,an input state ρˆin is
of one atom in the mode i, of two atoms in the modes firsttransformedintoasuperpositionoftwomodes,anal-
i and j, and of three atoms in the modes i, j, and k ogoustothetwoarmsofanopticalinterferometer. These
(with i,j,k = 1,2), respectively, and , denotes the modesacquiredistinctphasesϕ andϕ duringthe sub-
{· ·} 1 2
anti-commutator. To shorten notation we write the loss sequentquantumevolutionandarefinallyrecombinedto
rate of two (three) atoms in the same mode i as γi =γii read out interference fringes, from which the phase shift
(κi =κiii) and setκ12 =κ112 and κ21 =κ122. Note that ϕ = ϕ1 ϕ2 is inferred. We assume in the whole paper
−
theinter-moderatesγ12,κ12,andκ21 vanishforexternal thatduringthisinterferometricsequenceonecanneglect
BJJs. The loss rates depend on the macroscopic wave inter-particleinteractions(nonlineartermsintheHamil-
function of the condensate and thus on the number of tonian(3))andlossprocesses. Thisiswelljustifiedinthe
atoms and interaction energies Ui. As far as the number experiments of Ref. [13]. The dependence of the phase
of lost atoms at the revival time T remains small with sensitivity on inter-particle interactions has been stud-
respecttotheinitialatomnumberN0,onemay,however, ied in [34, 35]. Under this assumption, the output state
assumethattheseratesaretime-independentinthetime of the interferometer is ρˆout(ϕ) = e−iϕJˆn~ρˆineiϕJˆn~, where
interval [0,T]. Hereafter we always assume that this is Jˆ =n Jˆ +n Jˆ +n Jˆ is the angularmomentumgen-
the case. ~n x x y y z z
erating a rotation on the Bloch sphere along the axis
defined by the unit vector~n, with Jˆ =(aˆ†aˆ +aˆ†aˆ )/2,
x 1 2 2 1
4. Conditional states Jˆy =−i(aˆ†1aˆ2−aˆ†2aˆ1)/2, and Jˆz =(aˆ†1aˆ1−aˆ†2aˆ2)/2.
Thephaseshiftϕisdeterminedbymeansofastatisti-
cal estimator depending on the results of measurements
Themasterequation(11)doesnotcouplesectorswith
ontheoutputstateρˆ (ϕ). Thebestprecisiononϕthat
different numbers of atoms N. As a result, if the den- out
can be achieved (that is, optimizing over all possible es-
sitymatrixρˆ(t)hasinitiallynocoherencesbetweenstates
timators and measurements) is given by [23]
with different N’s then such coherences are absent at all
times t 0. Hence
1
≥
(∆ϕ) = , (14)
best
N0 F(ρˆ ,Jˆ )
in ~n
ρˆ(t)= ρ (t) , ρ (t)=w (t)ρˆ (t), (13) M
N N N N
q
NX=0 where is the number of measurements and
M
e e
wdehnesrietyρmNa(tt)rix(ρˆwNi(tth))aiwseltlh-deeufinnnedoramtoamlizendum(nboerrmNal(itzhedat) F(ρˆ,Jˆ )=2 (pk−pl)2 k Jˆ l 2 (15)
~n ~n
p +p h | | i
is,hn1,en2|ρN(t)|n′1,n′2i=0forn1+n2 6=N orn′1+n′2 6= k,l,pXk+pl>0 k l (cid:12) (cid:12)
N) and wN(t) 0 is the probability of finding N atoms (cid:12) (cid:12)
in the BJJ at t≥ime t (thus w (t) = 1). The matrix is the quantum Fisher information. Here, l is an
e N N {| i}
ρˆN(t)istheconditionalstatefollowingameasurementof orthonormal basis diagonalizing ρˆ, ρˆ|li = pl|li. The
Nˆ. Moreprecisely,itdescribPesthestateoftheBJJwhen quantum Fisher information thus measures the amount
of QCs in the input state that can be used to enhance
oneselectsamongmanysingle-runexperimentsthosefor
phase sensitivity with respect to the shot noise limit
which the measured atom number at time t is equal to
N and one averages all experimental results over these (∆ϕ) = 1/ Nˆ , that is, to the sensitivity ob-
SN
Mh i
“post-selected” single-run experiments, disregarding all tained by usinqg Nˆ independent atoms. Since Jˆ does
the others. In this sense, ρˆN(t) contains a more precise not couple subsphaceis with different N’s, it follow~ns from
physical information than the total density matrix ρˆ(t).
Eq.(15) and from the block structure (13) of ρˆthat
To have access to this information, one must be able to
extractsampleswithawell-definednumberofatomsini-
N0
tially (since we assumed an initial state with N0 atoms) F(ρˆ,Jˆ~n)= wNF(ρˆN,Jˆ~n), (16)
and after the evolution time t. Even though the precise
N=0
X
5
whereF (ρˆ ,Jˆ )isthe Fisherinformationofthecondi- jumps occur and the atomic state is transformed as
N N ~n
tionalstate ρˆ withN atomsandw isthe correspond-
N N
ing probability. Mˆm ψ(s )
ψ(s ) ψ(s+) = | − i , (19)
It is shown in [36] that if F(ρˆ,Jˆ~n) is larger than the | − i−→| i Mˆm ψ(s )
averagenumber of atoms Nˆ then the atoms are entan- k | − ik
gled. AccordingtoEq.(14)h,thieconditionF(ρˆ,Jˆ~n)> Nˆ where the index m labels the type of jump and Mˆm is
is a necessary and sufficient condition for sub-shot nhoisei thecorrespondingjumpoperator. Inourcase,restricting
sensitivity (∆ϕ) <(∆ϕ) . for the moment our attention to two-body losses, one
best SN
has three types of jumps: the loss of two atoms in the
InordertoobtainameasureofQCsindependentofthe
first mode, with Mˆ = aˆ2, the loss of two atoms in
direction~n of the interferometer, we optimize the Fisher 2,0 1
information over all unit vectors~n and define [37], the second mode, with Mˆ0,2 = aˆ22, and the loss of one
atom in each mode, with Mˆ = aˆ aˆ . The probability
1,1 1 2
F(ρˆ)= max F(ρˆ,Jˆ )=4C . (17) that a jump m occurs in the infinitesimal time interval
k~nk=1 ~n max [s,s+ds] is dp (s) = Γ Mˆ ψ(s) 2ds, where Γ is
m m m m
k | ik
the jump rate in the loss channel m. Using the notation
Here, Cmax is the largest eigenvalue of the 3 × 3 real of Sec. IIA3, one has Γ2,0 = γ1, Γ0,2 = γ2, and Γ1,1 =
symmetric covariance matrix γ . Between jumps, the wave function ψ(t) evolves
12
| i
according to the effective non self-adjoint Hamiltonian
1 (p p )2 Hˆ =Hˆ iDˆ with
C = k− l Re k Jˆ l l Jˆ k , eff 0− 2−body
ab a b
2 p +p h | | ih | | i
k,l,pXk+pl>0 k l (cid:8) (cid:9)(18) Dˆ2−body = 12 ΓmMˆm†Mˆm (20)
with a,b = 1,2,3. For simplicity we write Ftot(t) Xm
≡ 1 γ
F(ρˆ(t)) for the optimized Fisher information of the to- = γ nˆ (nˆ 1)+ 12nˆ nˆ .
i i i 1 2
tal atomic density matrix ρˆ(t) at time t (note that the 2 − 2
i=1,2
direction ~n maximizing F(ρˆ,Jˆ ) depends on t). When X
~n
The physicaloriginof the damping termcomes from the
studying the QCs in the conditional states we optimize
gainofinformationacquiredonthe atomic state by con-
over~n independently in each subspace and define F (t)
N
ditioning the system to have no loss in a given time in-
as in (17), by replacing ρˆby ρˆ (t) in this formula. Note
N
terval [39, 43]: the longer the time interval, the smaller
thatF (t) is notequalto w (t)F (t), because the
tot N N N mustbethenumberofatomsleftintheBJJinthemode
optimal directions may be different in each subspace.
P losing atoms.
In the absence of losses, the two-component superpo-
The random wave function at time t reads
sition of CSs has the highest possible Fisher information
F[ρˆ(0)(t2)] = N02, which is for N0 ≫ 1 approximately ψ (t) = |ψJ(t)i
twice larger than that of the superpositions with q com- J
| i ψ (t)
ponents, 3 q < N1/2 [18, 33]. The upper solid curve keJ k
in Fig. 2(a)≤sho∼ws F0tot(t) in the absence of losses as a |ψJ(t)i=e−ei(t−sJ)HˆeffMˆmJe−i(sJ−sJ−1)HˆeffMˆmJ−1···
function of time for N0 =10 atoms in the BJJ. e−iHˆeff(s2−s1)Mˆ e−is1Hˆeff ψ(0) , (21)
e ··· m1 | i
where J is the number of loss events in the time inter-
val [0,t], 0 s s t are the random loss
C. Quantum trajectories ≤ 1 ≤ ··· ≤ J ≤
times, and m ,...,m the random loss types. The time
1 J
evolution of the wave function t ψ (t) for a fixed
J
7→ | i
We solvethe master equation(11)using twomethods: realization of the jump process is called a quantum tra-
the quantum jump approach and an exact diagonaliza- jectory.
tion. We outline inthis sectionthe firstapproach,which The probability to have no atom loss between times 0
yields a tractable analytical solution in the case of few and t is given by e−itHˆeff ψ(0) 2. The probability to
loss events and gives physical intuition on the various haveJ losseventsikn[0,t],w|ithtihkeνtheventoftypem
ν
decoherence mechanisms. This approach will be used to occurringinthe timeinterval[s ,s +ds ],ν =1,...,J,
ν ν ν
explainthe resultsprovidedbythe exactdiagonalization is
method, which offers the exact solution for the whole
density matrix when inter-mode losses are absent, i.e. dp(mt)1,...,mJ(s1,...,sJ;J)
γ = κ = κ =0. The exact diagonalization method
12 21 12 =Γ ...Γ ψ (t) 2ds ...ds . (22)
isdescribedinAppendixA.Weuseitmostlytocompute m1 mJk J k 1 J
numerically the Fisher information. Thelinkofthisapproachwiththemasterequationde-
e
In the quantum jump description, the state of the scriptionisthattheaverageoverallquantumtrajectories
atoms is a pure state ψ(t) which evolves randomly in (that is, over the number of jumps J, the jump times
| i
time as follows [38–43]. At random times s quantum s , and the jump types m ) of the rank-one projector
ν ν
6
(a) symmetric case
100
(b) γ T =γ T =0.5
90 1 2
(i) no losses
80 (ii) γ T=0.025 5
1
70 (iii) γ1T=0.5 4 t2 20%
) 60 (iv) γ1T=5 t3
t N
(ot 50 F 3 43%
t
F 40 N
30 w 2 35% 8% 2%
20 1 28%43%
10 5%13%
0
0 0 2 4 6 8 10
0 0.1 t3 0.2 t2 0.3 0.4 0.5
N
t (T)
(c) all cases
50
(d) t=t =T/4
2
25
40
(i) sym. loss.
20 (ii) asym. loss.
t) 30 N (iv) asym. loss.
(ot F 15 and energies
t
F 20 N
10
w
10 (i) sym. loss. 5
(ii) asym. loss.
(iii) sym. loss., asym. energies
(iv) asym. loss. and energies 0
0 0 2 4 6 8 10
0 0.1 0.2 0.3 0.4 0.5
N
t (T)
FIG. 2. (Color online) (a) Total quantum Fisher information F (t) versus time t (in units of T = 2π/χ) for symmetric
tot
two-body loss rates γ = γ in each mode and γ = α = κ = κ = 0. The different curves correspond to (from top to
1 2 12 i i ij
bottom) γ T = 0, 0.025, 0.5, and 5. The dotted vertical lines indicate the formation times t = T/4 and t = T/6 of the 2-
1 2 3
and 3-component superpositions. The histogram (b) shows the contributions w F (ρˆ ,J ) to F of the subspaces with
N N N ~nopt tot
different atom numbers N [see Eq.(16)] for two different times, t (pink boxes) and t (blue boxes), and for the loss rates
2 3
indicated above the histogram. The percentages on top of each boxes are the probabilities w of finding N atoms at these
N
times (weights smaller than 1% are not indicated). (c) Same as in (a) for (i) symmetric losses (γ = γ = 0.177/T) and
1 2
energies (U = U ); (ii) asymmetric losses (γ = 0.6/T, γ = 0) and symmetric energies (U = U ); (iii) symmetric losses
1 2 1 2 1 2
(γ = γ = 0.177/T) and asymmetric energies (U = U < U ); (iv) asymmetric losses (γ = 0, γ = 0.6/T) and energies
1 2 2 12 1 1 2
(U =U <U ). The loss rates are chosen in such a way that the number of lost atoms at time t is the same and equal to
2 12 1 2
about3inallcases. TheinteractionenergiesU aresuchthatT =4π/(U +U −2U )isthesameinallcases. (d)Histogram
i 1 2 12
of thecontributionsof thesubspaces with N atoms toF (t ) for thesame valuesof γ and U asin (c) in thecases (i) (right
tot 2 i i
pink boxes), (ii) (middle green boxes), and (iv) (left purple boxes). In all panels N = 10, γ = 0, and one- and three-body
0 12
losses are absent. All results are obtained from theexact diagonalization method (see AppendixA).
ψ (t) ψ (t) yields the density matrix ρˆ(t) solution of depend on quantum trajectories having J two-body loss
J J
| ih |
the master equation(11)[39]. We thus recoverthe block events.
structure (13) of the atomic density matrix, with
It is straightforward to extend the above description
ρ (t)= Γ ...Γ ds ...ds to include also one- and three-body losses. This is
NJ m1 mJ 1 J
m1X,...,mJ Z0≤s1≤···≤sJ≤t aercahtioevrsedMˆby a=ddaˆin,gMˆnew=typaˆes(foofrjuonmep-bsowdiythlojsusmesp) aonpd-
e ψ (t) ψ (t) , (23) 1,0 1 0,1 2
| J ih J | Mˆ = aˆ3, Mˆ = aˆ3, Mˆ = aˆ2aˆ , and Mˆ = aˆ aˆ2
3,0 1 0,3 2 2,1 1 2 1,2 1 2
where we have set N = N 2J. Therefore, quantum (for three-body losses). The corresponding jump rates
e e J 0−
trajectories provide a natural and efficient tool to study are Γ = α , Γ = α , Γ = κ , Γ = κ ,
1,0 1 0,1 2 3,0 1 0,3 2
the conditionalstates ρˆ (t) with N atoms, whichonly Γ = κ , and Γ = κ . The conditional state
NJ J 2,1 12 1,2 21
7
ρˆ (t) is obtained by summing the right-hand side of thisfigure,inthepresenceoftwo-bodyasymmetriclosses
N
Eq.(23) over all J and all r ,...,r 1,2,3 such that only,the QCs of the superpositionsare wellpreservedas
1 J
∈{ }
N J r = N, r being the number of atoms lost in the small atomic sample discussed above. This asym-
in0t−heνtνh=l1osνsevent. TνheeffectiveHamiltonianbecomes metricsituationisrealizedintheexperimentofRef.[13],
Hˆ =PHˆ iDˆ with Dˆ =Dˆ +Dˆ +Dˆ the two-body losses occurring mainly in the upper inter-
eff 0 1−body 2−body 3−body
and − nal level [44]. When one- and three-body losses – which
are also present in this experiment – are added, the co-
1
Dˆ1−body = αinˆi (24) herencesofthesuperpositionsarestillpreservedprovided
2
i=1,2 that all losses occur in the second mode (upper energy
X
1 level). In this case, the QCs can be protected against
Dˆ = κ nˆ (nˆ 1)(nˆ 2)
3−body 2 i i i− i− atom losses by tuning the interaction energy U2 such
iX=1,2 that U2 = U12. This shows that the results of Ref. [21]
1 concerningtwo-body lossesholdfor one- and three-body
+ κ nˆ (nˆ 1)nˆ . (25)
ij i i j
2 − losses as well. However, when symmetric one-body or
i6=j
X three-body losses are added, the QCs are destroyedon a
much shorter time scale and the peak in the Fisher in-
III. MAIN RESULTS formation at time t2 disappears. In the experiments of
Refs. [12, 13], the one-body losses are symmetric since
they are due to collisions with atoms from the back-
We present in this section our main results on the
ground gas, which are equally likely for the two inter-
time evolution of the QCs in the atomic state under the
nal atomic states. These one-body symmetric losses are
quenched dynamics in the presence of atom losses. The
thereforemuchmore detrimentalto the QCs thanasym-
amountofQCsisestimatedbythe totalquantumFisher
metric two-body losses.
information F (t).
tot
Before investigating the combined effect of the vari-
ous loss processes, we start by a detailed analysis of a
smallatomicsamplewithN =10atomssubjecttotwo-
0
bodylossesonlyandwithoutinter-modelosses(thelatter
losses cannot be addressed by our exact diagonalization
and will be discussed in Sec. IV). Figure 2 shows the ef- 9000
(i) asym. 2body.
fectofincreasingthelossratesinasymmetricmodelwith 8000 (ii) asym. 123body
γ = γ and U = U (panel (a)). The Fisher informa- (iii) exp.
1 2 1 2
7000
tion, which in the absence of losses is characterized by a
broadpeakatthetimet =T/4offormationofthetwo- 6000
2
componentsuperposition,rapidlydecreasesoncetheloss )
t 5000
(
rate increases. If, however, an asymmetric model is cho- t
o
sen– withparametersyieldingthe sameaveragenumber Ft 4000
oflostatomsattimet andthesamerevivaltimeT asin 3000
2
thesymmetriccase–wefindthattheFisherinformation
2000
is considerably increased (panel (c) in Fig. 2). The most
1000
favorablesituationturnsouttobetheonewithasymmet-
ricenergiesU2 =U12 <U1andvanishinglossrateγ1 =0 0
0 50 100 150 200 250 300 350 400
in the mode with largest interactions (a similar result
t (ms)
would be obtained for U = U < U and γ = 0). The
1 12 2 2
histogramsshowninpanels(b)and(d)inFig.2givethe
FIG. 3. (Color online) Total quantum Fisher information
contributionsw (t)F(ρˆ (t),Jˆ )toF (t)ofthevarious
N N ~n tot F (t) versus time t (in units of T) from exact diagonal-
tot
subspaces with fixed atom numbers N (see (16)), evalu-
ization for more realistic experimental conditions (see Ap-
ated in the direction ~n = ~nopt optimizing the Fisher in- pendix B) with N0 = 100, U2 = U12, U1−U12 = 18.056Hz,
formationofthetotalstate,fort=t2,t3. Oneinfersfrom and(i)asymmetrictwo-bodylossesγ2=0.0127Hzandγ1 =0
these histograms that the aforementioned effect is non- withoutone-andthree-bodylosses;(ii)one-,two-,andthree-
trivial, namely, the large Fisher informations at times t2 body losses in the second mode with rates α2 = 0.4Hz,
and t3 for asymmetric rates and energies do not come γ2=0.0127Hz, κ2 =1.08×10−6Hz,andnolosses inthefirst
from the contribution of the subspace with N =N . mode; (iii) symmetric one- and three-body losses and asym-
0
metric two-body losses, α = α = 0.2Hz, γ = 0.0127Hz,
We study nowanatomic samplewith N =100atoms 1 2 2
0 γ =0, and κ =κ =0.54×10−6Hz. The case (iii) roughly
initially inthe casewhereseverallossprocessesarecom- 1 1 2
corresponds to theexperimental conditions in Refs. [12, 13].
binedtogether. Figure3showsthetotalquantumFisher
information for experimentally relevant parameters ex-
tracted fromRefs. [12, 14, 16] (we explainhow these pa-
rametersareobtainedinAppendix B).As canbeseenin
8
IV. QUANTUM CORRELATIONS IN THE finds
SUBSPACES WITH FIXED ATOM NUMBERS
1 N0 N
A. Overview of the results from the quantum jump wN0(t)=trρ(Nn0oloss)(t)= 2N0 n10 e−2tdN0(n1).
method nX1=0(cid:18) (cid:19)
(28)
e
Inthe followingwerestrictourattentiontosymmetric
In order to explain the behavior of the total Fisher three-bodylossesκ =κ andκ =κ . Theasymmetric
1 2 12 21
information observed in the exact diagonalization ap- three-bodylosscaseis treatedinAppendix C.Letus set
proach, we analyze separately the contributions of each κ=(3κ κ )/2 and
1 12
−
subspace with a fixed atom number to the total Fisher
1
information. This is done by using the quantum jump
a= γ +γ γ +(N 2)κ. (29)
1 2 12 0
approachof Sec. IIC. We argue in what follows that the 2 − −
stronger decoherence for symmetric two-body loss rates (cid:0) (cid:1)
Ifa=0,thedampingfactorinEq.(27)(i.e.theexponen-
and energies in Fig. 2(c) originates from a “destructive 6
tial factor in the right-hand side) is Gaussian. Actually,
interference” (exact cancellation) when adding the con-
by using Eqs.(20), (24), and (25), one obtains
tributions of the two loss channels at time t . A similar
2
lcoasnsceesl,labtuiotnitocicsuarbssaetnttimfoer to3nefo-brosdyymmloessterisc. thMroeere-boovedry, dN0(n1)=a(n1−n1)2+c, (30)
theweakerdecoherenceforcompletelyasymmetriclosses where c is an irrelevant n -independent constant which
1
obtained by tuning the interaction energies as described can be absorbed in the normalizationof the density ma-
above comes from the absence of dephasing in the mode trix, and
ilosingatoms. Thiseffectissomehowtrivialforexternal
1
BJJs: there this absence of dephasing occurs for a van- n = ∆α ∆γ+N (2γ γ )+2N (N 2)κ (31)
1 0 2 12 0 0
ishing interaction energy U = U = 0; for such U the 4a − − −
i 12 i (cid:16) (cid:17)
collision processes responsible of two-body losses in the with ∆α=α α and ∆γ =γ γ .
2 1 2 1
mode i are suppressed (moreover, our assumption that Inorderto e−stimate the QCs in−ρˆ (t) andthe typical
N0
thelossratesareindependentoftheenergiesisnotjusti- loss rates at which this state is affected by the Gaussian
fiedanymore). Incontrast,forinternalBJJsdecoherence damping, we now focus on three particular cases.
isreducedwhenUi isequaltotheinter-modeinteraction (i) Symmetric loss rates γ1 = γ2 and α1 = α2. In
Uby12in6=vo0kainngdtthheeeeffffeeccttivisenpohna-sterinvoiaisl.eWpreodshuacelldexbpylaaitnomit nthis=caNse/a2.=Hγe/n2c+e t(hNe0d−am2)pκinwgitfhactγor=in2γE1q−.(2γ712) aisnda
1 0
losses in the presence of interactions [21, 22]. Gaussiancentered at (n ,n′)=(N /2,N /2). This cen-
1 1 0 0
ter coincides with the peak of the matrix elements in
the absence of losses, which have a width √N , see
0
≈
B. Conditional state in the subspace with the Eq.(8). Thus the effect of the Gaussian damping begins
initial number of atoms to set in for times t such that at 1/N . In particu-
0
≈
lar, the macroscopic superposition at time t = π/(χq)
q
wiWthethstearintibtiyaldneutemrmbeinrinofgatthoemcsonNd0itiaotntailmsetatt.e ρIˆnN0t(hte) iiss snhootwicneaibnlyApapffeenctdeidx CbythdaatmρˆpNin0(gtqf)orcoanv>∼ergχeqs/aNt0l.argIet
quantum jump approach this corresponds to the contri- loss rates a χq to the pure Fock state N0/2,N0/2
≫ | i
butionofquantumtrajectorieswithno jump inthe time with equalnumbers ofatoms in eachmode if N0 is even,
interval [0,t], which are given by (see Sec. IIC) as it could have been expected from the symmetry of
the losses. This convergence is illustrated in the up-
ψ (t) =e−it(Hˆ0−iDˆ) N ;φ=0 . (26) per panels in Fig. 4, which represent the density matrix
0 0
| i | i (27) at time t = t for increasing symmetric two-body
3
lossratesandvanishingone-body,three-body,andinter-
The unnormealized conditional state is ρ(noloss)(t) =
|ψ0(t)ihψ0(t)|. In the Fock basis diagonalizNin0g both Hˆ0 mspoadcee rwaittehs.NT0haetFomishseartintfiomremta2tioisndFisNp0l(aty2e)dininthFeigs.ub5-.
and Dˆ, it takes the form e For γ = γ > 5/T, it is close to the Fisher information
1 2
e e F ( )=N∼(N /2+1) of the Fock state N /2,N /2 .
hn1,n2|ρ(Nn0oloss)(t)|n′1,n′2i= LeNt0u∞s,howev0er,0stressthatatsuchlossrate|sρˆ0N0(t2)0hais
a negligible contributionto the totaldensity matrix (13)
e−t[edN0(n1)+dN0(n′1)]hn1,n2|ρˆ(N00)(t)|n′1,n′2i, (27) and is very unlikely to show up in a single-run experi-
ment, because the no-jump probability w (t ) is very
where ρˆ(0)(t) is the state in the absence of losses (see small. Thus, the large value of F (t ) foNr0str2ong sym-
N0 N0 2
Sec. IIA2), n2 =N0−n1, n′2 =N0−n′1, and dN0(n1)= metric losses does not mean that the total atomic state
n ,n Dˆ n ,n . The probability to find N atoms at ρˆ(t) has a large amount of QCs. The Husimi distribu-
1 2 1 2 0
h | | i
time t is found with the help of Eqs.(7) and (27). One tions of the conditional state ρˆ (t ) are shown in the
N0 2
9
γ T =0 γ T =2 γ T =10 γ T =20
1 1 1 1
10 0.5 10 0.5 10 0.5 10 0.5
8 8 8 8
6 6 6 6
n’ 4 0.25 n’ 4 0.25 n’ 4 0.25 n’ 4 0.25
2 2 2 2
0 0 0 0
0 0 0 0
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
n n n n
γ T =0 γ T =4 γ T =20 γ T =40
1 1 1 1
10 0.5 10 0.5 10 0.5 10 0.5
8 8 8 8
6 6 6 6
n’ 4 0.25 n’ 4 0.25 n’ 4 0.25 n’ 4 0.25
2 2 2 2
0 0 0 0
0 0 0 0
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
n n n n
FIG. 4. (Color online) Moduli of the matrix elements in the Fock basis of the state ρˆ (t ) in the subspace with N atoms
N0 3 0
at the time of formation t = T/6 of the 3-component superposition, from the exact diagonalization method. Upper panels:
3
symmetric two-body losses (γ = γ ). Lower panels: completely asymmetric two-body losses (γ = 0). The values of γ are
1 2 2 1
indicated on the top of each panel. Otherparameters: α =γ =κ =κ =0 and N =10.
i 12 i ij 0
100 with the symmetric case, except for strongly asymmet-
θ symmetric ric one-body loss rates satisfying ∆α γ N . In the
90 asymmetric ≈ 1 0
φ last case, this onset occurs when γ1 χq/N0 as in case
80 θ (i). Therefore, if ∆α is not of the ≈order of γ1N0, the
θ Gaussiandampingaffectsmorestronglythemacroscopic
) 70 φ θ
(tN20 60 φ φ spuapneerlspoinsitFioign.s4threapnreisnentthethseymmamtreitxrieclecmaseen.tsTofheρˆNl0o(wt3e)r
F 50 inthe Fock basis andthe dashedcurvein Fig.5 displays
θ the Fisher information FN0(t2) for ∆α = κ = 0. Ex-
40
φ θ cept at small values of γ1, FN0(t2) is much smaller than
30 φ for symmetric losses. This can be explainedfrom the re-
sults of Appendix C, which show that ρˆ (t ) converges
20 N0 q
0 5 10 15 20 in the strongloss limit γ1 χq to the Fock state 0,N0
γ (1/T) if α2 < α1 and to a supe≫rposition of Fock state|s withi
1
n = 0 or 1 atoms in the first mode if α = α . These
1 1 2
pure states have Fisher informations of the order of N ,
0
FIG. 5. (Color online) Fisher information FN0(t2) optimized which are smaller by a factor of N0 than those obtained
inthesubspacewithN0 atomsattimet2 asafunctionofthe forstrongsymmetriclosses. Becausethe aforementioned
loss rate γ (in units of T−1). Solid line: symmetric losses
1 FockstatesarelocalizednearthesouthpoleoftheBloch
(γ = γ ). Dashed line: asymmetric losses (γ = 0). The
1 2 2 sphere (θ = 0), the two peaks in the Husimi functions
Husimi functions are plotted in the insets for some specific
(lower insets in Fig. 5) move to values of θ smaller than
choices of loss rates indicated by circles on the two curves.
π/2 when increasing γ . Note that this picture is drasti-
Otherparameters as in Fig. 4. 1
cally modified when α = γ N +α : then ρˆ (t ) con-
2 1 0 1 N0 2
vergestoasuperpositionofthe Fockstates N /2,N /2
0 0
| i
and N /2+1,N /2+1 for even N (see Appendix C),
0 0 0
upperinsetsinFig.5forvariousratesγ . Thetwopeaks | i
1 and thus F (t ) behaves like in the case (i).
at (θ,φ) = (π/2,0) and (π/2,π) of the two-component N0 2
superposition in the absence of losses are progressively (iii) Strong inter-mode two-body losses γ >γ +γ +
12 1 2
washed out at increasing γ1, until one reaches the φ- 2(N0 2)κ, i.e.a < 0. Then the onset of damping at
independent distribution of the Fock state |N0/2,N0/2i. time −tq occurs for |a| ≈ χq/N02, except when ∆γ ≈
(ii) Completely asymmetric two-body losses and no ∆α/(N 1), in which case it occurs for a χq/N .
0 0
− − | | ≈
three-body losses, γ = γ = κ = 0. Then a = γ /2 AsshownintheAppendixC,ρˆ (t )convergesatstrong
2 12 1 N0 q
and n = ∆α/(2γ ) + 1/2. The onset of the damp- losses either to the Fock state with n = 0 or n = N
1 1 1 1 0
ing on the q-component superposition is at the loss rate atoms in the first mode, which has a Fisher information
γ χq/N2,whichissmallerbyafactorofN compared F ( ) = N , or, if ∆γ = ∆α/(N 1), to the so-
1 ≈ 0 0 N0 ∞ 0 − 0 −
10
calledNOONstate,whichhasthehighestpossibleFisher Equation (33) means that, apart from damping effects
information FN0(∞)=N02. duetotheeffectiveHamiltonianHˆeff,atomlossescanbe
accounted for by random fluctuations of the two phases
θ and φ of the CS. For a single loss event (J =1), these
C. Conditional states with N <N0 atoms fluctuations have magnitude
1
We study in this subsection the contribution to the
δθ δs δ m , δφ =δs χ m
m m i i m m i i
total atomic density matrix ρˆ(t) of quantum trajectories ≃ 2
i=1,2 i=1,2
having J 1 jumps in the time interval [0,t]. (cid:12)X (cid:12) (cid:12)X (cid:12)
≥ (cid:12) (cid:12) (cid:12) (3(cid:12)7)
(we assume here δθ 1), where δs is the fluctua-
m m
≪
tion of the loss time s, whose distribution is given by
1. General results
Eq. (D14) in Appendix D. This analogy between atom
losses and φ-noise is already known in the literature in
We first fix some notation. Let t 7→ |ψJ(t)i be a the large N0 regime [22]. In this regime the θ-noise is
trajectory subject to J loss events, occurring at times negligible (see below).
0 ≤ s1 ≤ ··· ≤ sJ ≤ t. As in Sec. IIC we denote each The conditionalstates ρˆN(t) with N <N0 atoms turn
type ofloss by the pair m=(m1,m2) 1,2,3 2, where out to be simply related to the (unnormalized) density
∈{ }
m1 and m2 are the number of atoms lost in the first matrixconditionedtonolosseventforaninitialCSwith
and second modes, respectively. The associated jump N atoms, defined as follows
operator is Mˆ = aˆm1aˆm2. We use the vector notation
m 1 2
s = (s1,...,sJ) for the sequence of loss times sν and ρ(noloss)(t)=e−itHˆeff N;φ=0 N;φ=0eitHˆe†ff . (38)
m = (m ,...,m ) for the sequence of loss types m . N | ih |
1 J ν
Hloestreinmmν o=de(miνd,1u,rminνg,2t)hweiνththmloν,sistphreonceusms.beFrinoaflalyt,omlest Thee matrix elements of ρ(Nnoloss)(t) in the Fock basis are
given by Eq.(27) upon the replacement N N. It is
m = J (m +m ) be the total number of atoms 0 →
| | ν=1 ν,1 ν,2 demonstratedinAppendeixDthatifN <N0,thematrix
ejected from the condensate between times 0 and t.
ρˆ (t) is given in this basis by
P N
It is easy to see that each jump (19) transforms a CS
N ;θ,φ into a CS N r;θ,φ , where r = 1,2,3 is n ,n ρˆ (t)n′,n′
| 0 i | 0 − i h 1 2| N | 1 2i∝
the number of atoms lost during the jump. This CS is (t;n ,n′) n ,n ρ(noloss)(t)n′,n′ , (39)
rotated on the Bloch sphere by the evolution between EN 1 1 h 1 2| N | 1 2i
jTuhmispsrodtraitvieonnbisydthueentoontlhineedairffeeffreencttivneumHbamerisltoofnaiatonmHˆseffin. ownhetrheeEmN(att;rnix1,enn′1t)riiessannenavnedlopene′.deTpehnisdienngvoelnoptiemiesadned-
1 1
the BJJ in the time intervals [0,s1], [s1,s2], ,[sJ,t], termined explicitly in Appendix D. If a single r-body
···
leading to different interaction energies. More precisely, loss event occurs between times 0 and t, it is denoted by
it is shown in Appendix D that for three-body loss rates (1−jump)(t;n,n′) and is given by Eq.(D16). According
satisfying EN0−r
to Eq.(39), ρˆ (t) is given in the Fock basis by the loss-
N
κ ,κ (N t)−1 ,i,j =1,2,i=j , (32) lessdensitymatrixρˆ(0)(t)foraninitialCSwithN atoms
i ij 0 N
≪ 6
modulatedbytheenvelope andbythedampingfactor
N
the wavefunction ψ (t) is up to a normalizationfactor E
J of Eq.(27).
| i
given by
Letusassumethat, inadditiontothe abovecondition
|ψJ(t)i∝e−itHˆeff|N0−|m|;θm(s),φm(s)i, (33) (γ3i,2γ)1o2nthtr−ee1-.bFoudrythloesrsmeso,rteh,elettwtoh-ebotdoytallonssumrabteesrsmatisf=y
≪ | |
whereθm(s)andφm(s)arerandomanglesdependingon N0 N of atoms lost between times 0 and t be much
−
the loss types and loss times. These angles are given by smaller than N0. Then one finds (see Appendix D)
J 1
θm(s)=2arctan exp − s2ν δ1mν,1+δ2mν,2 EN(t;n,n′)= J1!J2!J3!
(cid:16) n Xν=1 (cid:0) (cid:1)o(cid:17) J1,J2,J3≥0,J1+X2J2+3J3=N0−N
φm(s)= J sν(χ1mν,1+χ2mν,2), (34) 3 EN(10−−jrump)(t;n,n′) Jr. (40)
νX=1 rY=1h i
where we have introduced the interaction energies Therefore, the envelope for several jumps is obtained by
multiplying together the single-jump envelopes raised to
χ =U U , χ = (U U ), (35)
1 1− 12 2 − 2− 12 the power Jr, and by summing over all the numbers Jr
and the loss rate differences of r-body losses in the time interval [0,t] such that N =
N J 2J 3J .
0 1 2 3
δ1 =2γ1 γ12+(3κ1 κ21)N0 , E−quati−ons (3−3), (39), and (40) are our main analyti-
− −
δ = (2γ γ +(3κ κ )N ). (36) cal results from the quantum trajectory approach. They
2 2 12 2 12 0
− − −