Single shot simulations of dynamic quantum many-body systems

3.1 MB·English

by  Kaspar Sakmann

Single shot simulations of dynamic quantum many-body systems
Kaspar Sakmann1 and Mark Kasevich1
1 Department of Physics, Stanford University, Stanford, California 94305, USA
The single-particle density is the most basic quantity that can be calculated from a
givenmany-bodywavefunction. Itprovidestheprobabilitytofindaparticleatagiven
position when the average over many realizations of an experiment is taken. However,
the outcome of single experimental shots of ultracold atom experiments is determined
5
by the N-particle probability density. This difference can lead to surprising results.
1
0 For example, independent Bose-Einstein condensates (BECs) with definite particle
2 numbers form interference fringes even though no fringes would be expected based
on the single-particle density [1–4]. By drawing random deviates from the N-particle
n
probability density single experimental shots can be simulated from first principles
a
[1, 3, 5]. However, obtaining expressions for the N-particle probability density of
J
realistic time-dependent many-body systems has so far been elusive. Here, we show
4
how single experimental shots of general ultracold bosonic systems can be simulated
1
basedonnumericalsolutionsofthemany-bodySchrödingerequation. Weshowhowfull

countingdistributionsofobservablesinvolvinganynumberofparticlescanbeobtained
]
s and how correlation functions of any order can be evaluated. As examples we show
a the appearance of interference fringes in interacting independent BECs, fluctuations
g in the collisions of strongly attractive BECs, the appearance of randomly fluctuating
- vortices in rotating systems and the center of mass fluctuations of attractive BECs
t
n in a harmonic trap. The method described is broadly applicable to bosonic many-
a body systems whose phenomenology is driven by information beyond what is typically
u
available in low-order correlation functions.
q
.
t
a Let us briefly outline how single experimental shots Methods for the algorithm and details.
m
can be simulated from a general many-body wave func-
- tion Ψ. The probability to find N particles at posi- 2
d tionsr ,...,r inamany-bodysystemisdeterminedby t = 5
n the N-1particleNprobability distribution P(r ,...,r ) = -2
co Ψ(r1,...,rN)2. In experiments on ultra1cold boNsons 2 t = 15
[ s|napshots of th|e positions of the particles are taken and -2

single experimental shots sample the N-particle proba- 2
1 t = 25
v bility distribution. This distribution is high-dimensional -2
4 andsamplingitdirectlyfromagivenN-bosonwavefunc- 2
2 tion is hopeless. However,it can be rewritten as a prod- t = 35
-2
2 uct of conditional probabilities
2
3
t = 45
0 P(r ,...,r )=P(r )P(r r ) P(r r ,...,r ), -2
1. 1 N 1 2| 1 ×···× N| N−1 (11) 2 y high
t = 55
0 where e.g. P(r2 r1) denotes the conditional probability -2 x low
15 to find a particle| at r2 given that another particle is at -20 0 20 -20 0 20
r . By drawing r from P(r), r from P(rr ), r from
v: P1(rr ,r ), etc., 1one random d2eviate of P|(1r ,..3.,r ) Figure1. Interferenceofindependentinteractingcondensates.
2 1 1 N
i | Two independent, repulsively interacting condensates collide
X is generated. Obtaining the conditional probabilities
in an elongated trap. Shown is the single-particle density
in (1) is a formidable combinatorial problem though,
r (left column) and random deviates of the N-particle density
a even for special cases [1, 5]. Here, we provide a gen-
(right column) at different times. In the overlap region in-
eral algorithm to simulate single shots from any given terference fringes show up in the N-particle density, but not
N-boson wave function |Ψi = ~nC~n|~ni, where |~ni = in the single-particle density. The results are obtained by
n1,...,nM areconfigurationsconstructedbydistribut- solving the many-body Schrödinger equation in two spatial
|ing N bosoins over M orbitals φP. We apply this algo- dimensions. Parameter values: N = 10000 bosons. Interac-
i
rithmtomany-bodystatesobtainedbysolvingthetime- tion strength λ = 4.95. See text for details. All quantities
dependent many-bodySchrödingerequationnumerically shown are dimensionless.
using the multiconfigurational time-dependent Hartree
for bosons method (MCTDHB) [6–8]. This combination ItisinstructivetobrieflyreviewBose-Einsteinconden-
ofmany-bodySchrödingerdynamicsandsamplingofthe sation. A many-boson state is condensed if its reduced
N-particleprobabilityallowsustosimulatesingleexper- single-particle density matrix has exactly one nonzero
imentalshotsfromfirstprinciplesinrealisticsettings,see eigenvalue ρ of order N [9]. The eigenvalues ρ are
i i
2
known as natural occupations, the eigenvectors as nat- 2 t=5 (a)t=25 (b) 1 (c)
ural orbitals. The BEC is fragmented if more than one
-2
eigenvalue ρ is of order N [10, 11], see Methods for de-
i 2 t=15 t=25
tails. Fully condensed states, i.e. states with ρ = N, N
are of the form φ(r1)φ(r2) φ(rN). (1) th1en be- -2 ρ(t)/i
comes a trivial product of in×de·p··e×ndent, identical proba- 2 t=25 t=25 ns
o
bilitydistributions,andtherearenocorrelationsbetween -2 ati
p
particles. Forinstance,Gross-Pitaevskii(GP)mean-field 2 t=35 t=25 u
c
c
states are of this form. Any other state, in particular -2 al o
fragmentedstates,exhibitcorrelationsandviceversaany 2 t=45 t=25 ur
correlated state is to some degree fragmented. We will Nat
-2
nowshowhowfragmentedBECsleadtomacroscopically
2 t=55 y t=25 high
fluctuating outcomes in single shots.
In the following we use dimensionless units ~ = m = -2 x low 0
1 and solve the time-dependent many-body Schrödinger -20 0 20 -10 0 10 0 10 Time 40 50
equation i∂ Ψ = Hˆ Ψ using the MCTDHB method
∂t| i | i Figure 2. Collision of independent attractively interacting
[6–8]. Here,
condensates. Two independent attractively interacting con-
densates collide in an elongated trap in two spatial dimen-
H = N 1 ∂2 +V(r )+λ δ (r r ) (2) sions. (a) Single-particle density at different times. The con-
−2∂r2 i 0 ǫ i− j densates approach each other without spreading and bounce
Xi=1 i Xi<j off one another. (b) Random deviates of the N-particle den-
sity at the time of the collision. Correlations lead to either
denotes a general many-body Hamiltonian in D dimen-
asinglestronglylocalized densitymaximumcontainingprac-
sions with an external potential V(r) and a regularized
tically all particles or two smaller maxima containing about
contact interaction δǫ(r) = (2πǫ2)−D/2e−r2/2ǫ2. We pa- half the particles each. (c) Fragmentation of the condensate
rameterizetheinteractionstrengthbythemean-fieldpa- asafunctionoftime. Theinitialstateistwofoldfragmented
rameter λ=λ0(N 1), see Methods for details. with ρ1/N = ρN/N =49.4%. During the collision two addi-
−
Let us begin with an example of two interfering, in- tionalnaturaloccupations becomesignificantly occupiedand
dependent condensates of N =10000 bosons in an elon- the system can no longer be separated into two independent
gated trap with tight harmonic confinement along the z condensates. Parameter values: N =100bosons. Interaction
strengthλ=−5.94. Seetextfordetails. Allquantitiesshown
directionsuchthatwecanworkinD =2dimensionsand
r = (x,y). We use V(r) = V (x)+V (y)+V (x) as an are dimensionless.
x y g
external potential, where V (x) and V (y) are harmonic
x y
traps and V (x) is an additional Gaussian potential that
g
flattens the bottomof the trapalongthe x-direction. As the same trap. For this purpose we use N =100 bosons
aninitialstateweusetwoindependentcondensates,each at an interaction strength λ = 5.94 which is about 2%
−
ofwhichisthemean-fieldgroundstate(correspondingto above the threshold for collapse of the GP mean-field
M =1intheMCTDHBformalism)ofN/2bosonsofthe ground state in this trap. For the initial state we first
displacedtrapsV (r)=V(x d,y)withd=18.6inhar- compute the many-body ground state of fifty bosons us-
±
monicoscillatorunits ofthe y±-directionataninteraction ing two orbitals and imaginary time-propagation. This
strength λ=4.95. The initial state Ψ(0) = N/2,N/2 ground state is highly condensed, ρ1/N = 98.7%. The
is fragmented with ρ = ρ = N|/2. iWe |then solvei initial state is then taken as the symmetrized product of
1 2
thetime-dependentmany-bodySchrödingerequationfor the ground state and a displaced copy of it located at
Ψ(0) using M = 2 orbitals. Fig. 1 shows the single- r = ( d,0). Thus, the initial state has natural occupa-
p|articile density as well as random deviates of the N- tions−ρ1/N = ρ2/N = 49.4% and ρ3/N = ρ4/N = 0.6%.
particle density at different times. The two condensates WethenpropagatethisinitialstateusingM =4orbitals.
acceleratetowardseachother,collideandseparateagain. Fig. 2 (a) shows the single particle-density at differ-
During the collision interference fringes appear in devi- enttimes. Thecondensatesapproacheachotherwithout
ates of the N-particle density at locations that fluctuate spreading significantly, collide and separate again. Dur-
randomlyfromshottoshot,butnotinthesingle-particle ing the collision the single-particle density exhibits two
density. Thisisalsoexpectedbasedonsimplifiedmodels maxima, the condensates seem to bounce off each other.
[1,3]. However,here this resultfollows directly fromthe However,singleshotsatthe time ofthe collisionreveala
solution of the many-body Schrödinger equation. The different result, see Fig. 2 (b). In about half of all shots
interparticle interaction is weak here; interaction effects a strongly localized density maximum is visible, whereas
only become visible as ripples in the density after the in the other half two smaller well separated maxima ap-
collision and the natural occupations remain practically pear. We stress that at no point any type of (possibly
constant all along. random) phase relationship between the colliding parts
We now go one step further and investigate collisions was assumed. In fact, for independent condensates the
between strongly attractive independent condensates in assumption of a preexisting, but random relative phase
3
1
(a) t=100 (b) mean-field
3 M=2
0 M=3
-3 800 M=10
N exact
3 t=200 ρ(t)/i
0 s 600
n s
-3 patio ount
3 t=300 cu C 400
c
0 al o
-3 ur
at 200
N
3 t=400
0 y
-3 high 0
x low
0 -10 -5 5 10
-3 0 3 -3 0 3 -3 0 3 -3 0 3 0 Time 400 Center of mass X
Figure 3. Fluctuating vortices. A repulsive condensate in Figure4. Fullcountingdistributionof thecenterofmassop-
the ground state of a harmonic trap is stirred by a rotating erator. Shown are 10000 random deviates of the center of
potential in two spatial dimensions. Over the course of time mass operator of the ground state of an attractively inter-
the system fragments and vortices appear at random posi- acting condensate in one spatial dimension. The center of
tionsinsingleshots. (a)Firstcolumn: single-particledensity mass fluctuations of the mean-field result (blue) are signifi-
at different times. Second to fourth column: single shots at cantlysmallerthanthoseofthemany-bodyresultswherethe
the same times. (b) Fragmentation of the condensate as a bosons are allowed to occupy M = 2,3,10 (green, magenta,
function of time. Starting from a condensed state, the sys- red) orbitals. The M = 10 result coincides with the exact
tem of bosons fragments as it is stirred. While the system analytical one (black). Parameter values: N = 10 bosons;
is condensed single shots and the single-particle density look interaction strength λ=−0.423, trap frequency ωx =1/100.
alike. Whenthesystemisfragmentedvorticesappearatran- All quantitiesshown are dimensionless.
dom positions. Parameter values: N = 10000. Interaction
strength: λ = 17. See text for details. All quantities shown
are dimensionless.
(b). While the system is condensed, single shots repro-
duce the single-particle density. Over the course of time
an additional natural orbital becomes occupied and the
is at variance with quantum mechanics [12]. The macro-
BEC becomes correlated. As correlations build up the
scopic fluctuations in the outcomes follow directly from
outcome of single shots fluctuates more and more and
the intrinsic correlations of the many-body state. Fig. 2
vorticesappear atrandomlocations in everysingle shot.
(c)showsthenaturaloccupationsofthesystem. Aslong
This is in stark contrastto mean-field theory, where due
asthecondensatesarefarapart,thenaturaloccupations
to the lack of correlations vortices always appear at the
remain close to their initial values. However, during the
same location.
collisiontwoadditionalnaturalorbitalsbecomeoccupied
indicating a buildup of even stronger correlations. As a As a last example let us show how full distribution
consequence after the collision the system can no longer functions of N-body operators can be evaluated by sim-
be separated into two independent condensates. ulating single shots. Consider the ground state of N at-
In the previoustwo examplesalreadythe initial states tractively interacting bosons in a harmonic trap, ωx =
were fragmented. We now turn to a system where frag- 1/100, in one dimension, i.e. D = 1 and r = x. The
mentation builds up dynamically. Stirring a BEC can exact wave function of the center of mass coordinate
lead to fragmentationand vortex nucleation that cannot X = N1 ixi of the many-body ground state is given
be explained within the mean-field framework of quan- by a GauPssian Ψmb(X) = (√πXmb)−1/2e−X2/2Xm2b with
tized vortices [5, 13]. Consider the ground state of a X = 1/√Nω [14]. On the other hand, the mean-
mb x
repulsively interacting BEC of N = 10000 bosons in a field ground state is uncorrelated and hence its center
pancake shaped trap with ω =ω =1 at an interaction of mass width is given by X = σ /√N, where
x y mf mf
strength λ = 17. We compute the many-body ground σ2 = φ x2 φ is the variance of the mean-field
mf h mf| | mfi
state using M =2 orbitalswhichis practicallyfully con- orbital φ , see Methods. In the limit of a weak trap,
mf
densed with ρ /N =99.98%. We then switch on a time- ω 0, the mean-field solution approaches a soliton
1 x
dependentstirringpotentialV (r,t)= 1η(t)[x(t)2 y(t)2] with→σ = π/(√3λ). Thus, for sufficiently strong at-
s 2 − mf | |
thatimpartsangularmomentumontotheBEC.Herex(t) tractive interaction X exceeds X . We compute the
mb mf
andy(t)varyharmonicallyandtheamplitude η(t)islin- groundstateofN =10bosonsataninteractionstrength
early ramped up from zero to a finite value and back λ= 0.423 using imaginary time-propagation for differ-
−
down,see Methods for details. Fig. 3 (a)shows the den- entnumbersoforbitals. Fromtheobtainedgroundstates
sity together with single shots at different times. The wegenerate10000randomdeviatesofthe centerofmass
evolution of the natural occupations is shown in Fig. 3 coordinate. Fig. 4 shows fits to the obtained histograms
4
of the center of mass deviates together with the exact C(k) ~n from the wave function Ψ(k−1) =
~n ~n | i | i
center of mass distribution. The many-body result for C(k−1) ~n , where the sums over run over all config-
M = 10 orbitals is indistinguishable from the exact one P~n ~n | i
urations of n and n+1 bosons, respectively. Defining
and significantly broader than the mean-field (M = 1) ~Pnq =(n ,...,n +1,...,n ) one finds from (4)
1 q M
result. In the present example the many-body correla-
tions are the cause for the onset of the delocalization of M
the ground state. C(k) = φ (r)C(k−1) n +1 (6)
~n Nk q ~nq q
q=1
X p
METHODS Using (6) in a general M orbital algorithm requires an
orderingofthe n+M−1 configurations ~n forallparticle
n | i
Bose-Einstein condensation. numbers n =1,...,N. Combinadics [7] provide such an
(cid:0) (cid:1)
ordering by associating the index
ForanN-bosonstate Ψ = C (t)~n andabosonic
fipealrdticolepedreantsoirtyΨˆm(ra)tri=x i|s dijeˆbfijnPφejd~n(ra)s~nthe| rieduced single- J(n1,...,nM)=1+Mi=−11(cid:18)n+M −1M−−i−i Pij=1nj(cid:19)
P X
(7)
ρ(1)(r|r′)=hΨ|Ψˆ†(r′)Ψˆ(r)|Ψi= ρijφ∗i(r′)φj(r) (3) witheachconfiguration|~ni. Using(7)allcoefficientsC~n(k)
i,j can then be obtained by evaluating the sums in (6) and
X
isdeterminedbynormalization. Usingthecoefficients
wtaiitnhs ρρi(j1)(=r|rh′Ψ) |=ˆb†iˆbj|Ψiρii.φNiBOy(rd)φiaNigOon∗(arli′z)i.ngThρeijeoigneenvoabl-- rCNk~n(k+k1) wfreomevaPlu(artrekρ,k.(.r.),ra1n)d. Tbyhimsecaonnscloufd(e5s)twheetahlgenorditrhamw
ues ρ1 ρ2 ... are known as naturaloccupations, the to simulate sin|gle shots. It is now easy to see that also
eigenve≥ctors≥φNi O(rP) as natural orbitals. If there is only correlationfunctionsofarbitraryordercanbeevaluated.
one eigenvalue ρ1 = (N) the BEC is condensed [9], if By realizing that
O
morethanoneρ = (N)theBECisfragmented[10,11].
i
The diagonal ρ(r) Oρ(1)(rr′ = r) is the single-particle k
density of the N-bo≡son wav|e function. ΨΨˆ†(r1)...Ψˆ†(rk)Ψˆ(rk)...Ψˆ(r1)Ψ = ρj−1(rj)
h | | i
j=1
Y
(8)
Single Shot Algorithm. the k-th order correlation function is evaluated at
r ,...,r as the product of the reduced densi-
1 k
Here we show how single shots can be simulated from ties ρj−1(rj). To evaluate the correlation function
a general N-boson wave function expanded in M or- ΨΨˆ†(r1)...Ψˆ†(rk)Ψˆ(rk)...Ψˆ(r1)Ψ the only modifi-
h | | i
bitals Ψ = C ~n , where ~n = n ,...,n and cation to the single shot algorithm above consists in
M n| i= N. ~nSp~nec|iail cases (f|ori M |=1 2) havMeibeen choosing the positions r1,...,rk rather than drawing
i=1 i P them randomly.
treated in earlier works [1, 5]. The goal is to draw the
Ppositionsr ,...,r ofN bosonsfromtheprobabilitydis-
1 N
tribution P(r ,...,r ). We achieve this by evaluating
1 N
MCTDHB.
the conditional probabilities in (1). For this purpose we
define reduced wave functions
In the MCTDHB [6–8] method the many-boson wave
Ψ , if k =0 function is expanded in all configurations that can be
Ψ(k) = | i (4)
| i (NkΨˆ(rk)|Ψ(k−1)i, if k =1,...,N −1 cdoenpsetnrduecntetdorbbyitadlisstφrib(ru,ttin).g NTheboasnosnastzovfoerr tMhe ttiimmee--
i
of n = N k bosons with normalization constants dependent many-boson wave function reads:
−
. The respective single-particle densities are given by
k
TρNkh(er)fi=rsthΨp(oks)i|tΨˆio†n(r)rΨˆ(irs)|dΨr(akw)inafnrdomNkP=(r)ρk=−1(ρrk()r−)/1N/2.. |Ψ(t)i= ~n C~n(t)|~n;ti (9)
1 0 X
Assuming that positions r ,...,r have already been
k 1 In (9) the C (t) are time-dependent expansion coeffi-
drawn, the conditional probability density for the next ~n
cientsandthe ~n;t aretime-dependentpermanentsbuilt
pisagrtivicelne bPy(r|rk,...,r1) = P(r,rk,...,r1)/P(rk,...,r1) from the orbit|als iφi(r,t). The MCTDHB equations of
motionarederivedbyrequiringstationarityofthemany-
P(rr ,...,r ) ρ (r), (5) body Schrödinger action functional
k 1 k
| ∝
S[ C (t) , φ (x,t) ]= dt Ψ(t)H i∂ Ψ(t)
since P(rk,...,r1) is a constant. The problem is { ~n }{ j } {h | − ∂t| i
thus reduced to obtaining the wave function |Ψ(k)i = − Mk,j=1µRkj(t)[hφk|φji−δkj]}, (10)
P
5
with respect to variations of the coefficients and the or- on a grid [ 43.2,43.2] [ 3.6,3.6]. For the rotating
− × −
bitals. The µ (t) are time-dependent Lagrange multi- BEC the parameter values are ω = ω = 1 and η(t)
kj x y
pliersthatensuretheorthonormalityofthe orbitals. For is linearly ramped up from zero to η = 0.1 over
max
bosons interacting via a delta-function interaction and a time span t = 80. η(t) is then kept constant for
r
M =1 the MCTDHB equations of motion reduce to the t = 220 and ramped back down to zero over a time
up
time-dependent Gross-Pitaevskii equation. For more in- span t . The potential V (r,t)= 1η(t)[x(t)2 y(t)2] ro-
r s 2 −
formation see the literature [6–8]. tates harmonically with x(t)=xcos(Ωt)+ysin(Ωt) and
y(t)= xsin(Ωt)+ycos(Ωt), where Ω = π/4. The grid
−
size is [ 8,8] [ 8,8].
− × −
For the D = 1 dimensional simulations we as-
Parameters. sume tight harmonic confinement along the y- and
z-directions with a radial frequency ω = ω = ω
⊥ y z
For the D = 2 dimensional simulations in this work and an oscillator length l⊥ = ~/(mω⊥). The contact
we assume tight harmonic confinement with a frequency interaction potential is then given by 2~2aδ (x), with
ω and a harmonic oscillator length l = ~/(mω ) p ml2⊥ ǫ
z z z δ (x) = (2πǫ2)−1/2e−x2/2ǫ2. We use ~ω as the unit of
along the z -direction. The bosons interact via a reg- ǫ ⊥
ularized contact interaction potential ~2λ0δp(r), with energy and l⊥ as the unit length. The dimensionless
m ǫ interaction strength is then given by λ = 2a/l . The
δ (r) = (2πǫ2)−1e−r2/2ǫ2 and a dimensionless interac- 0 ⊥
ǫ harmonic potential along the x-direction ω = 1/100
x
tion strength λ0 = √8πa/lz, where a is the scattering is much weaker than the radial confinement ω⊥ = 1.
length and m the mass of boson. We note that it is im- The grid size is [ 90,90]. The Gross-Pitaevskii soli-
portant to regularize contact interaction potentials for ton solution on a−n infinite line takes on the form
D > 1 [15, 16]. The contributions to the external po- φ (x)= λ/4sech(λx/2).
mf
tential are given by V (x) = 1mω2x2, V (y)= 1mω2y2,
x 2 x y 2 y
and V (x) = Ce−x2/2σ2, with C = mσ2ω2. We obtain p A. Image processing.
g x
dimensionless units ~ = m = 1 and the Hamiltonian (2)
by measuring energy in units of ~ωy, length in units of The histograms of the positions of particles obtained
l = ~/(mω ) and time in units of 1/ω . We use a using the single shot algorithm have a resolution that is
y y y
plane wave discrete variable representation to represent determinedby the gridspacing. For better visibility and
p
allorbitalsandoperators. Thewidthofthecontactinter- in analogy to a realistic imaging system we convoluted
actionisǫ=0.15andthegridspacingis∆x=∆y =ǫ/2 the data points of each histogram with a point-spread
for all simulations in this work. For the elongated trap function (PSF). As a PSF we used a Gaussian of width
the parameter values are ω = 0.07,ω = 1 and σ = 10 3 3 pixels.
x y
×
[1] Javanainen, J. & Yoo, S. M. Quantum Phase of a delberg (2013)
Bose-Einstein Condensate with an Arbitrary Numberof [9] Penrose, O. & Onsager, L. Bose-Einstein Condensation
Atoms. Phys.Rev.Lett. 76, 161 (1996) and Liquid Helium. Phys.Rev. 104, 576 (1956)
[2] Andrews, M. R. et al. Observation of Interference Be- [10] Nozières,P.&JamesD.S.Particlevs.paircondensation
tween Two Bose Condensates. Science 275, 637 (1997) inattractiveBose-liquids.J.Phys.France431133(1982)
[3] Castin, Y. & Dalibard, J. Relative phase of two Bose- [11] Streltsov, A. I. Alon, O. E. & Cederbaum, L. S. Gen-
Einstein condensates. Phys. Rev.A 55, 4330 (1997) eral variational many-body theory with complete self-
[4] Hofferberth, S. Lesanovsky, I. Fischer, B. Verdu, J. consistency for trapped bosonic systems. Phys. Rev. A
&SchmiedmayerJ.Radiofrequency-dressed-statepoten- 73, 063626 (2006)
tials for neutral atoms. Nat. Phys. 2, 710 (2006) [12] Mullin, W. J. & Laloë, F. Interference of Bose-Einstein
[5] Dagnino, D. Barberán N. & Lewenstein, M. Vortex nu- condensates: Quantumnonlocaleffects.Phys.Rev.A78,
cleation inamesoscopic Bosesuperfluidand breakingof 061605(R) (2008).
theparity symmetry.Phys.Rev. A 80, 053611 (2009) [13] Weiner,S.E.Tsatsos,M.C.Cederbaum,L.S.&Lode,A.
[6] Alon,O.E.Streltsov,A.I.&Cederbaum,L.S.Multicon- U.J.Angularmomentumin interactingmany-bodysys-
figurational time-dependent Hartree method for bosons: tems hides in phantom vortices. arXiv:1409.7670 (2014)
Many-body dynamics of bosonic systems. Phys. Rev. A [14] Brey, L. Johnson, N. F. & Halperin, B. I. Optical and
77, 033613 (2008) magneto-optical absorption in parabolic quantum wells.
[7] Streltsov, A. I. Alon, O. E. & Cederbaum, L. S. Gen- Phys. Rev.B 40, 10647 (1989)
eralmappingforbosonicandfermionicoperatorsinFock [15] Esry, B. D. & Greene, C. H. Validity of the shape-
space. Phys. Rev.A 81, 022124 (2010) independent approximation for Bose-Einstein conden-
[8] Streltsov, A. I. Sakmann, K. Lode, A. U. J. Alon, O. sates. Phys. Rev.A 60, 1451 (1999)
E. & Cederbaum L. S., The Multiconfigurational Time- [16] Doganov,R.A.Klaiman,S.Alon,O.E.Streltsov,A.I.&
Dependent Hartree forBosons Package,version2.3,Hei- Cederbaum,L.S.Twotrappedparticlesinteractingbya
6
finite-rangetwo-bodypotentialintwospatialdimensions. simulations. K. S. and M. K. wrote the paper.
Phys.Rev.A 87, 033631 (2013)
ACKNOWLEDGEMENTS COMPETING INTERESTS
FinancialsupportthroughtheKarelUrbanekPostdoc-
The authors declare that they have no competing fi-
toral Research Fellowship is gratefully acknowledged by
nancial interests.
K. S. Computing time was provided by the High Per-
formance Computing Center (HLRS) in Stuttgart, Ger-
many.
CORRESPONDING AUTHOR
CONTRIBUTIONS
Correspondence and requests for materials should
K. S. and M. K. conceived the ideas and designed the be addressed to Kaspar Sakmann (email: kas-
study. K.S.developedthealgorithmandcarriedoutthe [email protected])