Tunneling Dynamics in Open Ultracold Bosonic Systems: Numerically Exact Dynamics – Analytical Models – Control Schemes

143 Pages·2015·4.767 MB·English

by  Axel U. J. Lode (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Axel U. J. Lode
Tunneling Dynamics
in Open Ultracold
Bosonic Systems
Numerically Exact Dynamics,
Analytical Models, and
Control Schemes
Springer Theses
Recognizing Outstanding Ph.D. Research
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Axel U. J. Lode
Tunneling Dynamics
in Open Ultracold
Bosonic Systems
Numerically Exact Dynamics,
Analytical Models, and Control Schemes
Doctoral Thesis accepted by
the University of Heidelberg, Germany
123
Author Supervisor
Dr. AxelU.J.Lode Prof.Dres h.c. LorenzS.Cederbaum
Condensed Matter Theory Theoretical Chemistry Group
andQuantum ComputingGroup Universityof Heidelberg
Universityof Basel Heidelberg
Basel Germany
Switzerland
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‘‘... quantum physics means that anything
can happen at any time for no reason.’’
Professor Farnsworth, Futurama,
Bender’s Game
Supervisor’s Foreword
The tunneling effect lies at the very heart of quantum mechanics. Quantum par-
ticles can penetrate through potential barriers even if they lack the energy to
overcomethem.Classicalparticlescanonlypassoverthepotentialbarrierandnot
through it. There is hence no classical equivalent of the tunneling effect. The
amount of scientific work on tunneling of a single particle through a barrier is
enormous. There is, however, very little knowledge on how a system made of
severalandcertainlyofmanyparticlestunnelsthroughabarriertoopenspace.The
quantum mechanical process of tunneling in open many-body systems is offun-
damental interest for many different branches of science. This is simply because
almost all systems of interest in, for instance chemistry and physics, are open
many-body systems. In chemistry, tunneling occurs in photoassociation and pho-
todissociationprocesses,innuclearphysicstunnelingisimportantforalphadecay
and nuclear fission and fusion.
Inthetunnelingtoopenspace process, aparticle thathastunneledthrough the
barrierwillnotreturntothepotentialwelloutofwhichithassucceededtoescape.
If several particles have tunneled, the situation becomes more intricate as these
particles feel a drastically different situation outside the barrier than in the well.
That the particles outside the well are no longer confined and, moreover, inter-
acting with each other makes the mathematical and numerical treatment of tun-
neling to open space an extremely challenging task. Imagine that one particle has
tunneled and the second one follows only some time later. In the meantime, the
firstonemightalreadyhavetravelledalongdistance.Thisconsiderationmakesit
clear that one has to be able to treat a large if not enormous portion offree space
correctlyandpreciselyinordertodescribethetunnelingofasystemmadeofafew
and even more so of many particles to open space. The present thesis reports
numerically exact descriptions of up to N ¼1001 bosons in a spatial domain of
more than 7 mm long (!). This, for a quantum many-body system, is truly
tremendous.
The field of Bose–Einstein condensates has enjoyed enormous activity since
their first production with ultracold gases in 1995. This unique state of matter of
dilute atomic ultracold gases, is highly attractive for both experimental and
theoreticalscientistssincethestrengthoftheinteractionbetweenthebosons,their
dimensionality, as well as the shape of the trap potential holding them can be
vii
viii Supervisor’sForeword
varied and controlled essentially at will. It is due to these unique possibilities to
controlultracoldatomsthattheyarenowusedasso-calledquantumsimulatorsfor
awidevarietyofothersystemsinsolidstate,particle,andeveninastrophysics.In
this spirit, the present theoretical thesis relies on ultracold bosons to study and
analyzetheeffectofquantummany-bodytunnelingtoopenspacetheoretically.In
the field, much attention has been paid to the investigation of the static and
dynamic properties of Bose–Einstein condensates in traps and optical lattices.
Here, there are two popular standard theories that dominate the literature: The
famousGross–Pitaevskiimean-fieldtheoryandtheBose–Hubbardlatticemodelof
condensed-matterphysics.Itisnotoverestimatingtonotethattherearemorethan
a thousand manuscripts in the APS Journals dealing with the physics and prop-
erties of Bose–Einstein condensates as ‘‘seen’’ by these two theories. It has to be
stressed that these theories rely on model considerations, whereas the current
thesis presents numerically exact results of the full time-dependent Schrödinger
equation. In that, the present thesis investigates in great detail and beyond the
standard models the physics of tunneling to open space and finds and describes
fascinating collective many-body phenomena, such as the mechanism of the loss
ofcoherenceintheprocess.Itisfurthermorefoundinaperformedcomparisonthat
these fascinating many-body effects are not properly accounted for or even not
contained in the standard Bose–Hubbard and Gross–Pitaevskii theories.
Thisthesisisamongthefirstreportsofnumericallyexactcomputationsforthe
nonequilibrium quantum dynamics of interacting bosons in one and two spatial
dimensions. The scheme pursued for the investigation of the tunneling process is
asfollows:ABose–Einsteincondensateisinitiallytrappedinapotentialwelland
then allowed to tunnel through a potential barrier to open space. Two generic
scenariosareaddressed.Inthefirst,thebarrierisendingatthesameenergyasthe
bottom of the well and thus all bosons can decay by tunneling out from the well
(tunnelingwithout athreshold).Inthe second scenario, thebarrierisendingat an
energy higher than the bottom of the well such that one or more particles of the
condensate will not be able to tunnel and will have to stay in the well (tunneling
with a threshold). The particles of the condensate are interacting with each other
and the tunneling, of course, depends on this interaction. Both the spatial and
momentum evolution of the full wave function of the condensate are investigated
ingreatdetailandtheresultsarevisualizedbyinspectingtheone-particleandtwo-
particledensitiesandcorrelationfunctionsofthesystemasafunctionoftime,both
in real and momentum space. The results show extremely interesting physics not
anticipated before and contradicting the above popular models. In particular, the
tunneling mechanism found without a threshold can be viewed as if the bosons –
althoughcondensedinthewell–tunnelone-by-oneandlosetheircoherenceinthe
process of escaping, i.e., the system fragments in open space! The study of tun-
neling with a threshold reveals, using the above mechanism, a strategy which
allows a control of the momentum distribution and even the coherence properties
(!) of the escaping particles.
Supervisor’sForeword ix
Thenumericallyexactcalculationswerecarriedoutwithanimplementationof
themulticonfigurationaltime-dependentHartreeforbosons(MCTDHB)method–
the MCTDHB Package, see http://MCTDHB.org. The MCTDHB method is an
efficient many-boson wave-packet propagation technique for the nonequilibrium
dynamics of interacting bosons and is emerging as a leading many-body method
for the quantum dynamics of Bose–Einstein condensates. With such complicated
problems as the time-dependent many-boson Schrödinger equation, it is of great
importancetovalidateandbenchmarkthequalityofnewmethods.Thisisdonein
thisthesisbyidentifyingamodelsystemofinteractingbosonswhichcanbesolved
exactly and subsequently using the MCTDHB Package, and comparing the pre-
dictions of the latter against these exact solutions of the many-boson Schrödinger
equation. With this strategy, the present thesis first establishes the numerical
exactnessofMCTDHBformany-bosongroundstatesinoneandtwodimensions.
Subsequently and, most importantly, it is verified that MCTDHB can also
numerically exactly calculate the real-time propagation of many-boson systems
withageneralizedHamiltonianwithbothtime-dependentone-bodyandtwo-body
terms. The excellent agreement with the exact results is very valuable and
establishes the MCTDHB method as a new standard for the physics of
time-dependent many-boson systems.
Heidelberg, Germany, April 2014 Prof. Lorenz S. Cederbaum
Acknowledgments
My research in the recent years which culminates in this thesis was supported by
many people to whom I feel deeply indebted.
Firstofall,IwouldliketothankmysupervisorProf.Dr.LorenzS.Cederbaum.
HealwaysknewwhatobjectivetopursuewhenIwasatmywit’send.Theunique
working environment provided in the Theoretical Chemistry group is an effect of
hisapproachtoscience–withoutthisencouragingandmotivatingenvironmentas
well asthe brilliantway Lorenz Cederbaum asks scientific questions, I would not
have finished this thesis.
The scientific work presented in this thesis builds on intense and inspiring
discussionswith,mentoring,tutoring,andthecontinuoussupportbyProf.Dr.Ofir
E. Alon as well as Dr. Alexej I. Streltsov. I want to acknowledge also their and
Dr. Kaspar Sakmann’s contribution in copy-editing this thesis.
Next, Iwould like tothank Prof. Dr. Selim Jochim for pioneering experiments
on the tunneling dynamics in open few-particle systems which are closely related
tothephysicsofthepresentthesis.Furthermore,Iwanttothankhimforrefereeing
this thesis. Finally, I would like to acknowledge him and the members of his
Ultracold Quantum Gases group in Heidelberg for stimulating and interesting
discussions.
I feel indebted for discussions with, criticism, and proof-reading service of
Adrian Komainda, Alexander Kuleff, Alexej Streltsov, Samuel Markson, Ofir
Alon, Daniel Pelaez-Ruiz, Elke Faßhauer, Matthis Eroms, Shachar Klaiman,
Ulrike Lode, Marios Tsatsos, Shirin Faraji, Hans-Dieter Meyer, Lorenz S.
Cederbaum, Kaspar Sakmann, and Andreas Deuchert.
I would like to express my gratitude for the support and encouragement by
DimitriosR.M.D.K.Voularinos,DimitriosJ.J.I.Papakonstantinou,BjörnD.P.P.
Jessen, Efi Anestedi, Aliki Petratou, Marios Tsatsos, Daniel Pelaez-Ruiz, Elke
Faßhauer, Shirin Faraji, Susana Gomez, and the Therotical Chemistry group as a
whole.
MysiblingsUlrikeandHolgerLodeaswellasmyparentsMarianneandGernot
Lode are acknowledged for their unconditional support and encouragement.
Last but not least, I am indebted to the relentless and invaluable inspiration,
incentive,andsupportbySimoneLode(néeBruggesser)throughoutalltheyears.
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