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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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PHYSICS

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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.

1. Pitaevskii LP, Stringari S (2003) Bose–Einstein Condensation, (Oxford University Press,

Oxford).

2. Gaudin, M (1971) Boundary energy of a Bose gas in one dimension. Phys Rev A

4:386–394.

3. Cederbaum LS, Streltsov AI (2003) Best mean-field for condensates. Phys Lett A

318:564–569.

4. Gericke T, Würtz P, Reitz D, Langen T, Ott H (2008) High-resolution scanning electron

microscopy of an ultracold quantum gas. Nat Phys 4:949–953.

5. Serwane F, et al. (2011) Deterministic preparation of a tunable few-Fermion system.

Science 332:336–338.

6. Heine D, et al. (2010) A single-atom detector integrated on an atom chip: Fabrication,

characterization and application. New J Phys 12:095005.

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

Lode et al. www.pnas.org/cgi/doi/10.1073/pnas.1201345109 4 of 4