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How an interacting many-body system tunnels
through a potential barrier to open space
Axel U.J. Lodea,1, Alexej I. Streltsova, Kaspar Sakmanna, Ofir E. Alonb, and Lorenz S. Cederbauma
aTheoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany; and
bDepartment of Physics, University of Haifa at Oranim, Tivon 36006, Israel
Edited by Eric Johnson Heller, Harvard University, Cambridge, MA, and approved July 3, 2012 (received for review January 24, 2012)
The tunneling process in a many-body system is a phenomenon
which lies at the very heart of quantum mechanics. It appears in
nature in the form of α-decay, fusion and fission in nuclear physics,
and photoassociation and photodissociation in biology and chem-
istry. A detailed theoretical description of the decay process in
these systems is a very cumbersome problem, either because of
verycomplicatedorevenunknowninterparticle interactions ordue
to a large number of constituent particles. In this work, we theo-
retically study the phenomenon of quantum many-body tunneling
in a transparent and controllable physical system, an ultracold
atomic gas. We analyze a full, numerically exact many-body solu-
tion of the Schrödinger equation of a one-dimensional system with
repulsive interactions tunneling to open space. We show how the
emitted particles dissociate or fragment from the trapped and
coherent source of bosons: The overall many-particle decay process
is a quantum interference of single-particle tunneling processes
emerging from sources with different particle numbers taking
place simultaneously. The close relation to atom lasers and ioniza-
tion processes allows us to unveil the great relevance of many-
body correlations between the emitted and trapped fractions of
the wave function in the respective processes.
coherence ∣ cold atoms ∣ many-body physics ∣ quantum dynamics ∣
fragmentation
T
place in all systems whose potential exhibits classically forbidden
but energetically allowed regions. See, for example, the overview
in ref. 4 and Fig. 1. When the potential is unbound in one direc-
tion, the quantum nature of the systems allows them to overcome
potential barriers for which they classically would not have suffi-
cient energy and, as a result, a fraction of the many-particle sys-
tem is emitted to open space. For example, in fusion, fission,
photoassociation, and photodissociation processes, the energetics
or life times are of primary interest (5–9). The physical analysis
was made under the assumption that the correlation between de-
cay products (i.e., between the remaining and emitted fractions of
particles) can be neglected. However, it has to be stressed first
that, at any finite decay time, the remaining and emitted particles
still constitute one total many-body wave function and, therefore,
can be correlated. Second, in contrast to the tunneling of an
isolated single particle into open space, which has been amply
studied and understood (4), nearly nothing is known about the
tunneling of a many-body system. In the present study, we de-
monstrate that the fundamental many-body aspects of quantum
tunneling can be studied by monitoring the correlations and
coherence in ultracold atomic gases. For this purpose, the initial
state of an ultracold atomic gas of bosons is prepared coherently
in a parabolic trapping potential which is subsequently trans-
formed to an open shape allowing for tunneling (Fig. 1, Upper).
In this tunneling system the correlation between the remaining
and emitted particles can be monitored by measuring deviations
from the initial coherence of the wave function. The close rela-
tion to atom lasers and ionization processes allows us to predict
he tunneling process has been a matter of discussion (1–3)
since the advent of quantum mechanics. In principle, it takes
coherence properties of atom lasers and to propose the study of
ionization processes with tunneling ultracold bosons.
In the past decade, Bose–Einstein condensates (BECs)
(10, 11) have become a toolbox to study theoretical predictions
and phenomena experimentally with a high level of precision and
control (12). BECs are experimentally tunable, the interparticle
interactions (13), trap potentials (14), number of particles (15),
their statistics (16) and even dimensionality (17–19) are under
experimental control. From the theoretical point of view, the
evolution of an ultracold atomic cloud is governed by the time-
dependent many-particle Schrödinger equation (TDSE) (20)
with a known Hamiltonian (Methods). To study the decay scenario
in Fig. 1, we solve the TDSE numerically for long propagation
times. For this purpose, we use the multiconfigurational time-
dependent Hartree method for bosons (MCTDHB) (for details
and literature see Methods). Our protocol to study the tunneling
process of an initially parabolically trapped system into open
space is schematically depicted in Fig. 1, Upper. We restrict our
study here to the one-dimensional case, which can be achieved
experimentally by adjusting the transverse confinement appropri-
ately. As a first step, we consider N ¼ 2; 4, or 101 weakly repul-
sive
Methods for a detailed description of the considered experimental
parameters).
The initial one-particle density and trap profile are depicted in
Fig. 1, Upper. Next, the potential is abruptly switched to the open
form Vðx; t ≥ 0Þ indicated in this figure, allowing the many-boson
system to escape the trap by tunneling through the barrier
formed. Eventually, all the bosons escape by tunneling through
the barrier, because the potential supports no bound states.
The initial system, i.e., the source of the emitted bosons, is an
almost totally coherent state (21). The final state decays entirely
to open space to the right of the barrier, where the bosons
populate many many-body states, related to Lieb–Liniger states
(22–24), which are generally not coherent. It is instructive to ask
the following guiding questions: What happens in between these
two extremes of complete coherence and complete incoherence?
And how does the correlation (coherence) between the emitted
particles and the source evolve? By finding the answers to these
questions we will gain a deeper theoretical understanding of
many-body tunneling, which is of high relevance for future tech-
nologies and applied sciences. In particular, this knowledge will
allow us to determine whether the studied ultracold atomic
clouds qualify as candidates for atomic lasers (11, 25–27) or as
a toolbox for the study of ionization or decay processes (5–8).
Because the exact many-body wavefunctions are available at
any time in our numerical treatment, we can quantify and moni-
tor the evolution of the coherence and correlations of the whole
87Rb atoms in the ground state of a parabolic trap (see
Author contributions: A.U.J.L., A.I.S., K.S., O.E.A., and L.S.C. designed research, performed
research, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed. E-mail: axel.lode@pci.uni-heidelberg.de.
This article contains supporting information online at www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1201345109/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1201345109PNAS Early Edition
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system as well as between the constituting parts of the evolving
wave packets. In the further analysis we use the one-particle den-
sity in real ρðx; tÞ and momentum ρðk; tÞ spaces, their natural oc-
cupations ρNO
i
We defer the details on these quantities to Methods. To study the
correlation between the source and emitted bosons we decom-
pose the one-dimensional space into the internal IN and external
OUTregions with respect to the top of the barrier, as illustrated
by the red line and arrows in Fig. 1, Lower. This decomposition of
the one-dimensional Hilbert space into subspaces allows us to
quantify the tunneling process by measuring the amount Px
of particles remaining in the internal region in real space as a
function of time. In Fig. 2 we depict the corresponding quantities
for N ¼ 2 and N ¼ 101 by the green dotted curve. A first main
ðtÞ and correlation functions gð1Þðk0jk;tÞ (28–30).
not;ρðtÞ
observation is that the tunneling of bosonic systems to open space
resembles an exponential decay process.
The key features of the dynamics of quantum mechanical sys-
tems manifest themselves very often in characteristic momenta.
Therefore, it is worthwhile to compute and compare evolutions of
the momentum distributions ρðk; tÞ of our interacting bosonic
systems. Fig. 3 depicts ρðk; tÞ for N ¼ 2; 4; 101 bosons. At t ¼ 0
all the initial real space densities have Gaussian-shaped profiles
resting in the internal region (Fig. 1, Upper). Therefore, their
distributions in momentum space are also Gaussian-shaped and
centered around k ¼ 0. With time the bosons start to tunnel out
of the trap. This process manifests itself in the appearance of a
pronounced peak structure on top of the Gaussian-shaped back-
ground, see Fig. 3, Black Framed Upper. The peak structure is very
narrow (similar to a laser or an ionization process), the
bosons seem to be emitted with a very well-defined momentum.
For longer propagation times a larger fraction of bosons is
emitted and more intensity is transferred to the peak structure
from the Gaussian background. Thus, we can relate the growing
peak structures in the momentum distributions to the emitted
bosons and the Gaussian background to the bosons in the source.
We decompose each momentum distribution into a Gaussian
background and a peak structure to check the above relation
(Methods). The integrals over the Gaussian momentum back-
ground, Pk
close similarity of the Px
ticles remaining in the internal region in real space, and Pk
confirms our association of the Gaussian-shaped background
not;ρðtÞ, are depicted in Fig. 2 as a function of time. The
not;ρðtÞ, characterizing the amount of par-
not;ρðtÞ
Fig. 1.
(blue line) is prepared as the ground state of a parabolic trap Vðx; t < 0Þ
(dashed black line). The trap is transformed to the open shape Vðx; t ≥ 0Þ
(black line), which allows the system to tunnel to open space. (Lower)
Sequential mean-field scheme to model the tunneling processes. The bosons
are ejected from IN to OUTsubspaces (indicated by the red line). The chemical
potential μiis converted to kinetic energy Ekin;i. All the momenta correspond-
ing to the chemical potentials ki¼
appear in the momentum distribution, see Fig. 2. All quantities shown are
dimensionless.
Protocol of the tunneling process. (Upper) Generic density ρðx; t < 0Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2mEkin;i
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
2mμi
p
;i ¼ N; N − 1; …; 1
Fig. 2.
decay process. To confirm that the fraction of atoms remaining in the trap
decays exponentially with time, we depict the density-related nonescape
probabilities Px
the respective solid green symbols and red lines. All quantities shown are
dimensionless.
Many-body tunneling to open space is a fundamentally exponential
not;ρðtÞ in real and Pk
not;ρðtÞ in momentum space, indicated by
Fig. 3.
the physics of many-body tunneling to open space. The total momentum dis-
tributions ρðk; tÞ for N ¼ 101 (Black Framed Upper) and their peak structures
for N ¼ 2, N ¼ 4, N ¼ 101, and the respective Gross–Pitaevskii solutions, at
times t1< t2< t3< t4. The broad Gaussian-shaped backgrounds correspond
to the bosons remaining in the trap, the sharp peaks with positive momenta
can be associated with the emitted bosons. For N ¼ 2 we find two peaks in
Lower (i), for N ¼ 4 we find three peaks and an emerging fourth peak at
longer times in Lower (ii). Lower (iv) we find three washed out peaks for
N ¼ 101. The corresponding GP dynamics reveals only a single peak for all
times in Lower (iii). The arrows in the plots mark the momenta obtained from
the model consideration. All quantities shown are dimensionless.
The peak structures in the momentum distributions characterize
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with source bosons and the peak structure in the momentum dis-
tribution with emitted ones (Fig. 3, Black Framed Upper).
Now we are equipped to look into the mechanism of the many-
body physics in the tunneling process with a simplistic model.
When the first boson is emitted to open space one can estimate
its total energy as an energy difference, EN− EN−1, of source
systems made of N and N − 1 particles. Assuming negligible cou-
pling between the trapped and emitted bosons the energy of the
outgoing boson reads:
EOUT¼ EN− EN−1≡ μ1:
The quantity μ1is simply the chemical potential of the parabo-
lically trapped source system. For noninteracting particles its va-
lue would be independent of the number of bosons inside the
well. Because the boson tunnels to open space where the trapping
potential is zero, it can be considered for the time being as a
free particle with all its available energy, μ1, converted to kinetic
energy Ekin
have the momentum k1¼
momentum agrees excellently with the position of the peak in
the computed exact momentum distributions, see the arrows
marked k1in Fig. 3, Lower (i–iv)). This agreement allows us
to interpret the peak structures in ρðk; tÞ as the momenta of
the emitted bosons. As a striking feature, also other peaks with
smaller k appear in these spectra at later tunneling times (see
Fig. 3, Lower (i, ii, and iv)). We can use an analogous argumenta-
tion and associate the second peak with the emission of the sec-
ond boson and its chemical potential, μ2. Its kinetic energy and
the corresponding momentum, k2, can be estimated as the energy
difference μ2¼ EN−1− EN−2between the source subsystems
made of N − 1 and N − 2 bosons, under the assumption of zero
interaction between the emitted particles and the source. The
correctness of the applied logic can be verified from Fig. 3, Lower
(i and ii), where for the N ¼ 2 (N ¼ 4) particles the positions of
the estimated momenta k2(k3, k4) of the second (third, fourth)
emitted boson fit well with the position of the second (third,
fourth) peak in the computed spectra. The momenta and details
on the calculation are collected in the SI Text. The momentum
spectrum for N ¼ 101 bosons shows a similar behavior—the
multipeak structures gradually develop with time starting from
a single-peak to two-peaks and so on, see Fig. 3, Lower (iv).
However, from this figure we see that the positions of the peaks’
maxima, and with them the momenta of the emitted bosons,
change with time. On the one hand we see that the considered
tunneling bosonic systems can not be utilized as an atomic laser:
The initially coherent bosonic source emits particles with differ-
ent, weakly time-dependent momenta. In optics such a source
would be called polychromatic. On the other hand we can associ-
ate the peaks with different channels of an ionization process
and their time-dependency with the channels’ coupling. Thus,
we conclude that it is possible to model and investigate ionization
processes with tunneling ultracold bosonic systems. We mention
that such processes can occur in the multielectron photoioniza-
tion dynamics of molecules in a laser field. Namely, several elec-
trons can be ionized by tunneling through a barrier, see the recent
experiment in ref. 31.
Let us now investigate the coherence of the tunneling process
itself. In the above analysis ofthe momentum spectra we relied on
the exact numerical solutions of the TDSE for N ¼ 2; 4; 101
bosons. We remind the reader that in the context of ultracold
atoms the Gross–Pitaevskii (GP) theory is a popular and widely
used mean-field approximation describing systems under the
assumption that they stay fully coherent for all times. In our case
the GP approximation assumes that the ultracold atomic cloud
coherently emits the bosons to open space and keeps the source
and emitted bosons coherent all the time. To learn about the
coherence properties of the ongoing dynamics it is instructive
OUT¼k2
2m. We have to expect the first emitted boson to
ffiffiffiffiffiffiffiffiffiffiffiffi
2mμ1
p
. The value of the estimated
to compare exact many-body solutions of the TDSE with the
idealized GP results, see Fig. 3, Lower (iii). The strengths of
the interboson repulsion have deliberately been chosen such that
the GP gives identical dynamics for all N studied. It is clearly seen
that for short initial propagation times the dynamics are indeed
coherent. The respective momentum spectra obtained at the
many-body and GP levels are very similar, see Fig. 3, Lower
(i–iv) for ρðk; t1¼ 100Þ. At longer propagation times (t > t1),
however, the spectra become considerably different. This differ-
ence means that with time, the process of emission of bosons
becomes less coherent.
Next, to quantify the coherence and correlations between the
source and emitted bosons we compute and plot the momentum
correlation functions jgð1Þðk0; kjtÞj2in Fig. 4 (Left) for N ¼ 101
(for N ¼ 2; 4 they look almost the same). Let us stress here that
the proper correlation properties cannot be accounted for by ap-
proximate methods. For example the GP solution of the problem
gives jgð1Þj2¼ 1, i.e., full coherence for all times. For the exact
solution we also obtain that at t ¼ 0 the system is fully coherent,
and thus jgð1Þðk0jk;t ¼ 0Þj2¼ 1. Hence, Fig. 4, Upper Left is also
a plot for the GP time evolution. However, during the tunneling
process the many-body evolution of the system becomes incoher-
ent, i.e., jgð1Þj2→ 0. The coherence is lost only in the momentum-
space domain where the momentum distributions are peaked,
the k-region associated with the emitted bosons (Fig. 4, Left).
In the remainder of k-space the wave function stays coherent
for all times. We conclude that the trapped bosons within the
source remain coherent. The emitted bosons become incoherent
with their source and among each other. Therefore, the coher-
ence between the source and the emitted bosons is lost. A com-
plementary argumentation with the normalized real-space
correlation functions is deferred to SI Text.
In the spirit of the seminal work of Penrose and Onsager on
reduced density matrices (29), we tackle the following question:
how strong is the loss of coherence in many-body systems? The
natural occupation numbers, ρNO
i
the reduced one-body density (Methods) define how much the
system can be described by single, two, or more quantum mechan-
ical one-particle states. The system is condensed and coherent
when only one natural occupation is macroscopic and it is frag-
mented when several of the ρNO
i
(Right) we plot the evolution of the natural occupation numbers
for the studied systems as a function of propagation time. The
initial system is totally coherent—all the bosons reside in one
natural orbital. However, when some fraction of the bosons is
emitted, a second natural orbital gradually becomes occupied.
ðtÞ, obtained by diagonalizing
ðtÞ are macroscopic. In Fig. 4
Fig. 4.
lation functions in momentum space jgð1Þðk0jk;tÞj2for N ¼ 101 are plotted at
t ¼ 0, 400, 600, 700. At t ¼ 0 the system is totally coherent, i.e., jgð1Þj2¼ 1. At
times t > 0, the system remains coherent everywhere in k-space apart from
the region around k ¼ 1, where we find peaks in the momentum distribu-
tions. The loss of coherence, jgð1Þj2≈ 0 only in these regions allows us to con-
clude that the source (trapped) bosons remain coherent at all times whereas
the emitted ones are incoherent. (Right) The time evolution of the first few
natural occupation numbers ρNO
i
ðtÞ for N ¼ 2 (red lines), N ¼ 4 (green lines),
and N ¼ 101 (blue lines) bosons. The coherence in the systems is graduallylost
with time. The systems fragment because more and more natural orbitals
become populated. All quantities shown are dimensionless.
Monitoring the coherence of the system. (Left) The first-order corre-
Lode et al. PNAS Early Edition
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The decaying systems lose their coherence and become twofold
fragmented. For longer propagation times more natural orbitals
start to be populated, indicating that the decaying systems be-
come even more fragmented, i.e., less coherent. In the few-boson
cases, N ¼ 2 and N ¼ 4, one can observe several stages of the
development of fragmentation (32, 33)—beginning with a single
condensate and evolving toward the limiting form of N entangled
fragments in the end which means a full fermionization of the
emitted particles. In this fermionization-like case each particle
will propagate with its own momentum. For larger N the number
of fragments also increases with tunneling time. The details of the
evolution depend on the strength of the interparticle interaction
and number of particles. By resolving the peak structure in the
momentum spectra of the tunneling systems at different times
we can directly detect and quantify the evolution of coherence,
correlations, and fragmentation.
Finally, we tackle the intricate question whether the bosons
are emitted one by one or several at a time? By comparing the
momentum spectra ρðk; tÞ depicted in Fig. 3, Lower (i, ii, and iv)
at different times it becomes evident that the respective peaks
appear in the spectra sequentially with time, starting from the
most energetical one. If multiboson (two or more boson) tunnel-
ing processes would participate in the dynamics, they would give
spectral features with higher momenta which are not observed in
the computed spectra depicted in Fig. 3. A detailed discussion
and a model are given in SI Text. This model suggests that the
bosons tunnel out one by one. However, the fact that the peaks’
heights and positions evolve with time indicates that the indivi-
dual tunneling processes interfere, i.e., they are not independent.
The origin behind this interference is the interaction between
the bosons. We conclude that the overall decay by tunneling pro-
cess is of a many-body nature and is formed by the interference
of different single-particle tunneling processes taking place simul-
taneously.
We arrive at the following physical picture of the tunneling to
open spaceof an interacting, initially coherent bosonic cloud. The
emission from the bosonic source is a continuous, polychromatic
many-body process accompanied by a loss of coherence, i.e., frag-
mentation. The dynamics can be considered as a superposition of
individual single-particle tunneling processes of source systems
with different particle numbers. On the one hand, ultracold
weakly interacting bosonic clouds tunneling to open space can
serve as an atomic laser, i.e., emit bosons coherently, but only for
a short time. For longer tunneling times the emitted particles be-
come incoherent. They lose their coherence with the source and
among each other. On the other hand, we have shown the usage
of tunneling ultracold atoms to study the dynamics of ionization
processes. Each peak in the momentum spectrum is associated
with the single particle decay of a bosonic source made of N,
N − 1, N − 2, etc. particles—in close analogy to sequential single
ionization processes. These N discrete momenta comprise a total
spectrum—in close analogy to total ionization spectra.
We have focused here on the many-body tunneling process of
an initially coherent bosonic cloud. Nevertheless, it is interesting
to inquire whether the many-body tunneling mechanism would
change when the interaction between the bosons is increased,
e.g., by employing a Feshbach resonance. To this end, we have
repeated our studies for N ¼ 2; 4; 101 bosons with sevenfold
stronger interactions for which the initial state is still close to con-
densed, as well as for N ¼ 2; 4 200-fold stronger interacting
bosons, for which the initial state is already fermionized. The re-
sults are shown in SI Text. The tunneling mechanism is not altered
by the stronger interactions and our model predicts the positions
of the momentum peaks of the escaping bosons well.
As an experimental protocol for a straightforward detection of
the kinetic energy of the emitted particles one can use the pres-
ently available single-atom detection techniques on atom chips
(34) or the idea of mass spectrometry (see discussion in SI Text).
Summarizing, in many-particle systems decaying by tunneling to
open space the correlation dynamics between the source and
emitted parts lead to clearly observable spectral features which
are of great physical relevance.
Methods
Hamiltonian and Units. The one-dimensional N-boson Hamiltonian reads
as follows:
N
?
Here λ0> 0 is the repulsive interparticle interaction strength proportional to
the s-wave scattering length asof the bosons and xiis the coordinate of the
i-th boson. Throughout this work λ ¼ λ0ðN − 1Þ ¼ 0.3 for all considered N is
used (further results for N ¼ 2; 4; 101 bosons with sevenfold stronger inter-
action, λ ¼ 2.1, and for N ¼ 2; 4 bosons with 200-fold stronger interaction,
λ ¼ 60, are discussed at the end of the main text and SI Text). For convenience
we work with the dimensionless quantities defined by dividing the dimen-
sional Hamiltonian byℏ2
m is the mass of a boson, and L is a chosen length scale. At t < 0 trap is para-
bolic Vðx; t < 0Þ ¼1
in ref. 35.
In this work we consider87Rb atoms for which m ¼ 1.44316 · 10−25kg and
as¼ 90.4a0
0.0529 · 10−9m is Bohr’s radius. We emulate a quasi one-dimensional cigar-
shaped trap in which the transverse confinement is w⊥¼ 2; 291.25 Hz, which
is amenable to current experimental setups. Following ref. 36, the transverse
confinement renormalizes the interaction strength. Combining all the above,
the length scale is given by L ¼
mL2
ℏ¼ 1.37 · 10−3s. The relation between the (dimensionless) interaction
parameter λ0and the (dimension-full) scattering length asis hence given
by λ0¼2mω⊥L
^Hðx1; …; xNÞ ¼∑
j¼1
−1
2
∂2
∂x2
j
þ VðxjÞ
?
þ∑
j<k
N
λ0δðxj− xkÞ:
mL2, where ℏ ¼ 1.05457 · 10−34m2kg
secis Planck’s constant,
2x2, the analytic form of Vðx; tÞ after the opening is given
without tuning byaFeshbach resonance,where
a0¼
ℏλ0
2mω⊥as¼ 1.0 · 10−6m, and the timescale by
ℏ
· as.
The Multiconfigurational Time-Dependent Hartree for Bosons Method. The
time-dependent many-boson wave function ΨðtÞ solving the many-boson
Schrödinger equation i∂ΨðtÞ
∂t
and 38. Applications include unique intriguing many-boson physics such as
the death of attractive soliton trains (39), formation of fragmented many-
body states (40), and numerically exact double-well dynamics (41–43). Recent
optimizations of the MCTDHB, see, e.g., refs. 44 and 45, allow now for the
application of the algorithm to open systems with very large grids (here
216¼ 65;536 basis functions), a particle number of up to N ¼ 101, and an
arbitrary number of natural orbitals (here up to 14). We would like to stress
that even nowadays such kind of time-dependent computations are very
challenging.
The mean-field wave function is obtained by solving the time-dependent
Gross–Pitaevskii equation, which is contained as a special single-orbital case
in the MCTDHB equations of motion, see refs. 37 and 38. To ensure that the
tunneling wave packets do not reach the box borders for all presented
propagation times the simulations were done in a box ½−5;7465?. In the
dimensional units we thus solve a quantum mechanical problem numerically
exactly in a spatial domain extending over 8.29 mm.
¼^HΨðtÞ is obtained by the MCTDHB, see refs. 37
Many-Body Analysis of the Wave Function. With the many-boson wave func-
tion ΨðtÞ at hand the various quantities of interest are computed and utilized
to analyze the evolution in time of the Bose system. The reduced one-body
density matrix of the system is given by ρð1Þðxjx0;tÞ ¼ hΨðtÞj^Ψ†ðx0Þ^ΨðxÞjΨðtÞi,
where^Ψ†ðxÞ is the usual bosonic field operator creating a boson at position x.
Diagonalizing ρð1Þðxjx0;tÞ one gets the natural orbitals (eigenfunctions), ϕNO
and natural occupation numbers ρNO
i
ρð1Þðxjx0;tÞ ¼ ∑M
extent to which the system is condensed (one macroscopic eigenvalue)
or fragmented (two or more macroscopic eigenvalues) (32, 33, 46). The
diagonal part of the reduced one-body density matrix ρðx; tÞ ≡ ρð1Þðxjx0;tÞ
is the system’s density. The first-order correlation function in coordinate
space gð1Þðx0; x;tÞ ≡
ρðx;tÞρðx0;tÞ
of the interacting system (28, 30). The respective quantities in momentum
space, such as the momentum distribution ρðk; tÞ and the first-order
correlation function in momentum space gð1Þðk0jk;tÞ ≡
rived from ρð1Þðxjx0;tÞ via an application of a Fourier transform on its eigen-
functions.
In real space the density-related nonescape probability is given by
Px
i
,
(eigenvalues) from the expression
ðx0; tÞÞ?ϕNO
i¼1ρNO
i
ðtÞðϕNO
ii
ðx; tÞ. The latter determine the
ρð1Þðxjx0;tÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffip
quantifies the degree of spatial coherence
ρð1Þðkjk0;tÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρðk;tÞρðk0;tÞ
p
, are de-
not;ρðtÞ ¼ ∫INρðx; tÞdx (see ref. 35 for the non-Hermitian results). The
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momentum-density related nonescape probability Pk
least-squares fitting a Gaussian function ρGaussðk; tÞ ¼ Ae−ðBxÞ2to ρðk; tÞ in
the k-space domain ½−∞; 0?. A and B are the fit parameters. We then define
the momentum-density-related nonescape probability as
Z
not;ρðtÞ is obtained by
Pk
not;ρðtÞ ¼
ρGaussðk; tÞdk:
ACKNOWLEDGMENTS. We thank Shachar Klaiman and Julian Grond for a
careful reading of the manuscript and comments as well as Lincoln Carr for
discussions. ComputationtimeonthebwGRiDand theCray XE6cluster Hermit
at the High Performance Computing Center Stuttgart (HLRS), and financial
support by the Heidelberg Graduate School of Mathematical and Computa-
tional Methods for the Sciences (HGS MathComp) and the Deutsche For-
schungsgemeinschaft (DFG) also within the framework of the Enable fund
of theexcellence initiative at Heidelberg university are greatly acknowledged.
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PHYSICS
Page 6
Supporting Information
Lode et al. 10.1073/pnas.1201345109
SI Text
TheMany-BodyPhysicsofTunnelingtoOpenSpace.Let uscommence
this Supplementary Information by a model of the physical
processes constituting the many-boson tunneling of an initially
coherent bosonic system to open space. The processes are sche-
matically depicted in Fig. 1.
Static picture: Basic processes assembling the many-body physics.The
initial and final physical situations in the IN and OUTsubspaces
are intuitively clear. The totally condensed initial state lives in the
IN region and is confined by a harmonic potential. Therefore, it
can be described by a harmonic oscillator-like product state, see,
e.g., ref. 1. In the final state all the bosons have tunneled out and
live entirely in the semiinfinite OUTregion. According to ref. 2,
the static many-body solution of the one-dimensional bosonic sys-
tem with short-range repulsive interaction on a semiinfinite axis
can be constructed as a linear combination of many correlated
(incoherent) states. Therefore, the dynamical final state of our
system is incoherent.
To model the steps translating the fully coherent systems to
complete incoherence, let us first consider the situation in which
exactly one boson has tunneled through the barrier from IN to
OUT and has no more connection with the interior. The IN-
system now has N − 1 particles and the OUT-system has 1 par-
ticle. By assuming that no excitations have been produced in the
IN-system, the trapped bosons’ energy is exactly reduced by the
chemical potential μ1¼ EN− EN−1,see Fig. 1, Lower.Here Eiis
the energy of the trapped harmonic oscillator product state with
the distribution of i bosons in the IN subspace. We assume that
the chemical potential does not depend on the number of emitted
bosons, because in OUT VðxÞ ≈ 0. Let us further ignore the in-
terparticle interaction in the exterior system. Energy conservation
requires then that the chemical potential μ1of the first boson
tunneled from IN to OUT region must be converted to kinetic
energy. A free particle has the kinetic energy Ekin
we thus expect the first emitted boson to have the momentum
k1¼
that the many-body wave function can be considered in a loca-
lized basis jIN;OUTi. The process of emission of the first boson
in this basis reads jN;0i → jN − 1;1k1i. Here the k1superscript
indicates that the emitted boson occupies a state which is very
similar to a plane wave with momentum k1in the OUTsubspace.
Now we can prescribe the process of the emission of the second
boson as jN − 1;1k1i → jN − 2;1k1; 1k2i. By neglecting the inter-
actions between the first and second emitted bosons we can
define the second chemical potential as μ2¼ EINðN − 1Þ−
EINðN − 2Þ. Thus, a second kinetic energy Ekin
rise to the momentum peak at k2¼
mical potentials of the systems made of N − i and N − i − 1,
i ¼ 0; …; N − 1, particles are different, so the corresponding
peaks should appear at different positions in the momentum
spectra. We can continue to apply the above scheme until the last
boson is emitted j1;1k1⋯1kN−1i → j0;1k1⋯1kNi. Fig. 1, Lower in-
dicates the chemical potentials for these one-particle mean-field
processes by horizontal lines and the processes by the vertical
arrows. This simplified mean-field picture of the tunneling
dynamics is analogous to N sequential processes of ionization,
where the energetics of each independent process (channel) are
defined by the chemical potential of the respective sources made
of N, N − 1, N − 2, etc. particles.
OUT¼k2
2m—
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2mEkin
OUT;1
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
2mμ1
p
. The above considerations imply
OUT;2¼ μ2gives
. Generally, the che-
ffiffiffiffiffiffiffiffiffiffiffiffi
2mμ2
p
Connection of the model to the numerical experiment. Let us first
compute the momenta available in the system of N ¼ 2 bosons
with interparticle interaction strength λ0¼ 0.3, following ref. 3.
The difference between the total energies of the trapped system
made of N ¼ 2 and N ¼ 1 bosons provides k1¼ 1.106. The sec-
ond momentum associated with the emission of the last boson
from the parabolic trap gives k2¼ 1.000. A similar analysis done
for the system of N ¼ 4 bosons with the interparticle interaction
strength λ0¼ 0.1 [λ0ðN − 1Þ ¼ 0.3] gives k1¼ 1.106, k2¼ 1.075,
k3¼ 1.038 and k4¼ 1.000 for the first, second, third, and fourth
momentum, respectively. To relate the model and the full many-
body results we mark the momenta estimated from the respective
chemical potentials in Fig. 3 by vertical arrows. The agreement
between the momenta obtained from the model and the respec-
tive ones from the dynamics is very good, see the arrows and the
peaks in the orange framed plots in Fig. 3, Lower (i and ii). From
this figure it is clearly seen that the later in time we look at the
momentum distributions ρðk; tÞ, the closer the peaks’ maxima
locate to the estimated results. Moreover, our model explains
why for N ¼ 101 the peaks are washed out. The chemical poten-
tials of neighboring systems made of a big particle number (101
and 100) become very close and, as a result, the corresponding
peaks start to overlap and become blurred. Nevertheless, they are
always enclosed by the first and last chemical potentials contri-
buting, see the labels k1and kNin Fig. 3, Lower (iv).
The good agreement between our model and full numerical
experiments validates the applicability of the emerged physical
picture to the tunneling to open space. We continue by excluding
the possibility that the observed peaks in the momentum spectra
can be associated with excitations inside the initial parabolic trap
potential. Therefore, we calculate the chemical potentials asso-
ciated with the configurations where one or several bosons reside
in the second, third, etc. excited orbitals of the trapped system. It
is easy to demonstrate that the bosons emitted from these excited
orbitals would have higher kinetic energies resulting in spectral
features with higher momenta. Because the computed spectra
depicted in Fig. 3 do not reveal such spectral features we con-
clude that the excitations inside the initial parabolic trap poten-
tial do not contribute to the tunneling process in a visible manner.
The above analysis suggests that the overall many-body tunnel-
ing to open space process is assembled by the elementary mean-
field-like tunneling processes analogous to the ionization of the
systems made of different particle numbers which are happening
simultaneously. We also are in the position to deduce now that
every elementary contributing process is of a single-particle type.
Indeed, if it were a two-particle process, the kinetic energy of the
emitted bosons would have been Ekin
one can assume that the chemical potentials of the first two
processes are almost equal, i.e., μ2b
associated with a two-particle tunneling process would be
k2b
tot¼
the peaks occur in the exact solutions.
2b¼ðk2b
1Þ2þðk2b
2·2m
2Þ2
. For large N
1≈ μ2b
2≈ 2μ1. The momentum
ffiffiffiffiffiffiffiffiffiffiffiffi
4mμ1
p
¼
ffiffiffi
2
p
k1—which is far out of the domain where
TracingtheCoherenceinRealSpace.Here we complement our study
of the coherence in momentum space given in the manuscript by
its real space counterpart. To characterize the coherence of the
tunneling many-boson system in real space we compute the nor-
malized real space first-order correlation function gð1Þðx0
various times t for the system of N ¼ 101 bosons and depict the
results in Fig. S1. From this figure we see that initially the system
is fully coherent, namely jgð1Þðx0
jgð1Þðx0
1jx1;tÞ at
1jx1;t ¼ 0Þj2¼ 1. For t > 0
1jx1;tÞj2< 1 only in the OUT region, indicating that only
Lode et al. www.pnas.org/cgi/doi/10.1073/pnas.12013451091 of 4
Page 7
the emitted bosons quickly lose their coherence. In contrast, the
source bosons living in the interior around x1¼ x0
coherent for all times. This analysis of the real space correlation
function corroborates our findings from the first-order normal-
ized momentum correlation function gð1Þðk0
the main text: The bosons are ejected incoherently from a source,
which preserves its initial coherence.
1¼ 0 remain
1jk1;tÞ analyzed in
The Case of StrongerInteraction.In the main text we focused on the
many-body tunneling process of an initially coherent weakly in-
teracting bosonic cloud. In this section we show the generality of
the found mechanism of many-body tunneling by first analyzing
what happens in the case of sevenfold stronger interactions,
λ ¼ 2.1, when the initial state is still mostly condensed, but exhi-
bits larger depletion. We show the nonescape probabilities
Px∕k
case of theweak λ ¼ 0.3 interactions. Thereal spacequantity Px
is also very close to the momentum space quantity Pk
tigate the mechanism of the decay, we plot the momentum
distributions for N ¼ 2; 4; 101 bosons and the respective time-de-
pendent Gross–Pitaevskii calculation in Fig. S3 in the same way
as in Fig. 3. Our model predicts the characteristic momenta of the
dynamics well also in the case of sevenfold stronger interactions,
see the black arrows in Fig. S3. Of course, because the interaction
is stronger, the positions of the peaks in the momentum distribu-
tions shift to higher values, compare Fig. S3 and Fig. 3. Even,
when we turn to the case of 200-fold stronger interactions, where
the initial state is fermionized, the model predicts well the occur-
ring momenta in the momentum distributions as shown in Fig. S4
and the black arrows therein. To conclude, this analysis shows the
generality of the found mechanism of many-body tunneling to
open space as illustrated in Fig. 1.
notðtÞ for N ¼ 2; 101 in Fig. S2. The decay is faster than in the
not
not. To inves-
Direct Detection of the Momentum Spectra. It remains to line out
the possible straightforward experimental verification of the
emerged physical picture. In typical experiments the bosons are
ultracold many-electron atoms in a very well-defined electronic
state. According to the conjectures put forward above, the bosons
will tunnel to open space with definite kinetic energy. We propose
to detect the kinetic energy of the emitted bosons by utilizing the
techniques and principles of mass spectrometry as schematically
depicted in Fig. S5. One can place an ionization chamber at some
distance from the trapping potential to ionize the propagating
bosonic atom suddenly. The respective experimental ionization
techniques are presently available, see, e.g., ref. 4 and references
therein. The now charged particle will, by application of a static
electric field, experience a corresponding driving force and
change its trajectory. The trajectory of the ionized atom or, alter-
natively, the trajectory of the ionized electron are completely
described by the respective driving force, the electronic state of
the atom and its initial kinetic energy. By using a detector capable
to detect the charged atom or a photoelectron multiplier for the
electrons one can monitor the deflection of the ionized particle
from the initial direction of propagation. The kinetic energy and,
therefore, the momentum of the emitted boson can be calculated.
This procedure allows the in situ detection of the momentum
spectra ρðk; tÞ corresponding to different tunneling times and the
study of tunneling to open space as a function of time.
For the few-particle case it is especially interesting not only to
obtain the momentum spectra, but also to monitor the time
ordering in which the peaks appear, i.e., to monitor the time
evolution of the momentum peak densities ρðk; tÞ. In such an
experiment one can see whether the signals corresponding to
the different ki, i ¼ 1; …; N, will be detected sequentially, start-
ing from the largest momentum, or they appear to some degree
arbitrarily. The latter case is a clear indication that the tunneling
is a combination of several single particle tunneling processes
happening simultaneously, as we predict. Additionally, this
measurement would be among the first direct observations of the
dynamics of the coherence and normalized correlations in ultra-
cold bosonic systems.
Let us summarize. The deterministic preparation of few par-
ticle ultracold systems is now possible, see ref. 5. Mass spectro-
metry is one of the most well-studied techniques and working
tools available and even more sophisticated detection schemes
have been developed on atom chips (6). The combination of
these facilities makes the detailed experimental time-dependent
study of the tunneling mechanism feasible at present time.
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Oxford).
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Lode et al. www.pnas.org/cgi/doi/10.1073/pnas.12013451092 of 4
Page 8
Fig. S1.
decaying system of N ¼ 101bosons at various tunnelingtimes. White corresponds tojgð1Þj2¼ 1and black to jgð1Þj2¼ 0. The red lines inthe top left separate the
IN and OUTregions. Here white corresponds to full coherence and black to complete incoherence. In the OUTregion the spatial coherence is lost with time, i.e.,
jgð1Þj2≈ 0 on the off-diagonal jgð1Þðx0
discussion.
The real-space normalized correlation function of the tunneling to open space process. jgð1Þðx0
1jx1;tÞj2is used to measure the spatial coherence in the
1≠ x1jx1;tÞj2. The coherence of the source bosons is conserved, because in the IN part jgð1Þj2¼ 1 for all times. See text for
Fig. S2.
nentially with time, we depict the density related nonescape probabilities Px
green symbols and red lines. We conclude that even for stronger interactions the many-body tunneling to open space is a fundamentally exponential decay
process.
Same as Fig. 2 of the manuscript but for sevenfold stronger interaction. To confirm that the fraction of atoms remaining in the trap decays expo-
not;ρðtÞ in real and Pk
not;ρðtÞ in momentum space, indicated by the respective solid
Lode et al. www.pnas.org/cgi/doi/10.1073/pnas.1201345109 3 of 4
Page 9
Fig. S3.
and their peak structures for N ¼ 2, N ¼ 4, N ¼ 101, and the respective Gross–Pitaevskii solutions. The arrows in the plots mark the momenta obtained from the
model consideration. We conclude that even for stronger interactions the peak structures in the momentum distributions characterize the physics of many-
body tunneling to open space.
Same as Fig. 3 of the manuscript but for sevenfold stronger interaction. The total momentum distributions ρðk; tÞ for N ¼ 101 (Black Framed Upper)
Fig. S4.
mentum distributions’ peak structures for the N ¼ 2 and N ¼ 4 bosons with λ ¼ 60 (the color code is as in Fig. S3). The arrows in the plots mark the momenta
obtained from the model consideration.
The peak structures in the momentum distributions characterize the physics of strongly-interacting bosons tunneling to open space. The total mo-
Fig. S5.
from the experiment (Left) the bosons are ionized by, e.g., a laser beam (Left Middle). Subsequently, the ions/electrons are deflected by a static electric field and
counted by a detector (Right Middle). The momentum distribution can be obtained as histogram from different realizations of the few- or many-boson tunnel-
ing process by detection of the deflected particles (Right).
Proposed experimental realization ofthe momentum spectroscopy of the many-boson system tunneling to open space. At some propagation distance
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