Table Of ContentSpringer Theses
Recognizing Outstanding Ph.D. Research
Kazuma Nagao
Fluctuations and Non-
Equilibrium Phenomena
in Strongly-Correlated
Ultracold Atoms
Springer Theses
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Kazuma Nagao
Fluctuations
and Non-Equilibrium
Phenomena
in Strongly-Correlated
Ultracold Atoms
Doctoral Thesis accepted by
Kyoto University, Sakyo-ku, Kyoto
123
Author Supervisors
Dr. Kazuma Nagao Prof. KeisukeTotsuka
Institute for Laser Physics andCenter Yukawa Institute for TheoreticalPhysics
for Optical Quantum Technologies KyotoUniversity
University of Hamburg Sakyo-ku,Kyoto, Japan
Hamburg,Germany
Prof. Ippei Danshita
Department ofPhysics
KindaiUniversity
Osaka,Japan
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-981-15-7170-1 ISBN978-981-15-7171-8 (eBook)
https://doi.org/10.1007/978-981-15-7171-8
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To my parents.
’
Supervisor s Foreword I
High controllability andcleannessofultracold gases pavea waytoexperimentally
observing many interesting condensed-matter phenomena which are hard to
observeinother(say,solid-state)settings.Onepossibleapplication,whoseoriginal
idea may be traced back to the Feynman’s proposal in 1982, is to take this
advantage of cold-gas systems to simulate quantum many-body systems with
well-controlled experimental systems instead of using classical computers which
are suffering from various problems. Armed with state-of-the-art read-out tech-
niquessuchasquantum-gasmicroscope,thisideaofusingcoldgasesas“quantum
simulators” for strongly-correlated many-body systems, which are hard to tackle
with traditional theoretical approaches, has been successfully applied to various
equilibrium phenomena, e.g., superfluid-insulator transitions in correlated Bose
systems, antiferromagnetism in the two-dimensional Hubbard model, etc. On the
other hand, quantum simulation of out-of-equilibrium problems is far less devel-
oped mainly due to the lack of precise and reliable theoretical results to compare
with experiments. In this Thesis, Dr. Kazuma Nagao tries to fill the gap between
theory and experiments in quantum simulation of quantum dynamics.
Amongmanychallengingproblemsinthisresearcharea,Kazumaaddressestwo
important questions in his Thesis. The first is related to table-top simulation of the
Higgs (amplitude) mode with ultra-cold bosons in an optical lattice. Recent theo-
retical studies showed that quantum/thermal fluctuations and the existence of
non-uniform trap potentials obscure the Higgs mode in two dimensions, thereby
hindering the observation of the sharp peak in experiments. To explore the possi-
bility of observing clear experimental signatures of the Higgs mode in higher
dimensions, Kazuma investigates in Chap. 4 a particular response function of a
three-dimensional Bose system to obtain positive answers that the Higgs mode
survives fluctuations and can be seen in experiments even in the presence of
inhomogeneity as far as we carefully perturb the system. I believe that these
observations will motivate further “Higgs hunting” experiments in higher-
dimensional Bose systems in the near future.
vii
viii Supervisor’sForewordI
The second project is to develop a reliable theoretical tool that can be used to
simulate non-equilibrium phenomena in higher-dimensional correlated many-body
systems. Except for one-dimensional systems in which the time-dependent
density-matrix renormalization group method is applicable, there are only a lim-
ited number of numerical tools available to reliably simulate far-from-equilibrium
dynamics,suchasquantumquenches,ofquantummany-bodysystemstotheextent
that we can directly compare the data with experiments. To tackle this problem,
Kazumachosethesemi-classicalphase-spacemethodcalledtheTruncatedWigner
Approximation (TWA). The essence of the TWA, roughly speaking, is to replace
the complicated quantum-mechanical average of operators with that over classical
trajectories in the phase space. He spent some time in learning this relatively new
methodfromthescratchandvisitedthelaboratoryofprofessorPolkovnikovseveral
times to polish his skills and extend the method to fermions in the
strongly-correlatedregime (unfortunately,theoutcomeofthisinteresting projectis
not included in this Thesis). His effort results in a concise but very pedagogical
introductiontothemethodgiveninChap.3,which,togetherwithaquicksummary
of ultra-cold Bose gases in Chap. 2, is definitely very useful to those who are
interested in non-equilibrium phenomena in cold gases. The crowning Chap. 5 is
thehighlightofthisThesis,inwhichKazumatriestodemonstratetheutilityofthe
TWA approach in far-from-equilibrium dynamics. Specifically, he extends the
TWA to investigate various dynamical phenomena subsequent to a quantum
quench in two- and three-dimensional Bose-Hubbard systems, and finds a good
agreementwiththeexperimentaldata.Althoughweneedfurtherbenchmarktestsin
order to establish the TWA method as the standard tool, his results, I believe,
strongly suggest that the TWA is a promising method in far-from-equilibrium
problems and will contribute significantly to the development of quantum simu-
lation of non-equilibrium phenomena.
Kyoto, Japan Prof. Keisuke Totsuka
March 2020
’
Supervisor s Foreword II
Recent years have witnessed a rapid growth in studies of quantum many-body
physics thanks to technological advances in building and utilizing synthetic
quantum platforms, such as superconducting circuits, quantum dots, trapped ions,
Rydberg atoms trapped in an optical tweezer array, and ultracold gases in optical
lattices. Among those platforms, a system of an optical lattice loaded with an
ultracold gas especially has a long successful history in the application for analog
quantumsimulations.Sincethefirstobservationofthesuperfluid-to-Mottinsulator
transition of a Bose gas in an optical lattice in 2002, optical-lattice quantum sim-
ulators have been applied for studying various kinds of quantum many-body
physics so that they now attract much attention from a broad range of subfields of
physics, including not only atomic, molecular, and optical physics, and condensed
matter physics, but also nuclear and high-energy physics.
A drawback of such a rapidly growing topic with a long history is that hurdles
fornoviceresearchersarenowratherhighbecausetheaccumulatedknowledgeand
theavailabletoolboxarediverse,andthestate-of-theartresearchhasbecomemore
andmorecomplicated.ThisthesiswrittenbyDr.KazumaNagaocertainlyservesas
a useful textbook for such novices to learn basics of quantum many-body physics
of the optical-lattice systems. The high readability of this thesis can be attributed
partlytothefactthatitfocusesonasingle-componentBosegasinanopticallattice,
whichservesasaquantumsimulatorofthesingle-componentBose-Hubbardmodel
and is one of the simplest settings in various quantum simulators realizable with
opticallattices.AremarkableadvantageoftheBose-Hubbardmodelisthatmanyof
its important features in and near equilibrium, such as quantum phase transitions,
thermodynamics, and elementary excitations, can be captured through analytical
calculations at least qualitatively. In Chap. 2, Dr. Nagao carefully reviews those
analytical calculations with several different approaches so that the readers can
acquire deep insights into the system and proficiency in field-theoretical methods.
In Chap. 4, he further develops the analytical approaches by combining the local
density approximation with three standard field theoretical techniques, namely the
Schwinger boson representation, the Holstein-Primakov expansion, and the
finite-temperature Green’s function techniques. He predicts that the Higgs
ix
x Supervisor’sForewordII
amplitudemodeofBosegasesinathree-dimensionalopticallatticecanbedetected
as a resonance peak at accessible finite temperatures in the presence of a trapping
potential,demonstratingthatthebasicsofthequantummany-bodytheorypresented
in Chap. 2 are still very useful in the front-line research topic.
Another unique feature of this thesis is that Chap. 5 constitutes an exemplary
case of how theoretical researchers can use optical-lattice quantum simulators.
During his Ph.D. student period, Dr. Nagao engaged as a theoretical research
assistant in a project regarding development of optical-lattice quantum simulators
builtbytheQuantumOpticsGroupatKyotoUniversity.Inaspecificthemeofthe
project, the experimental group performed a quantum simulation of
non-equilibrium dynamics of the three-dimensional Bose-Hubbard model after a
sudden quench of theoptical-lattice depth starting witha Mottinsulating state toa
deep superfluid region. They measured the time evolution of the kinetic and
interaction energies. Since this high-dimensional system consisted roughly of ten
thousand particles, it was obviously infeasible for classical computers to simulate
the system in a numerically exact manner. Nevertheless, the accuracy of outputs
from the quantum simulator was ensured via thorough comparisons with exact
numerical computations in 1D. Hence, Dr. Nagao decided to utilize the quantum
simulation results as a quantitative reference for examining the performance of
some approximate numerical methods. In Chap. 5, he specifically chose the trun-
cated Wigner approximation based on the Gross-Pitaevskii equation, whose ped-
agogical review is presented in Chap. 3, in order to substantiate its quantitative
validity on the quench dynamics. He further applied the method with “a stamp of
approval” by the quantum simulator to the problem of spatial spreading of a
two-point correlation function and presented discussions from a viewpoint of
quasi-particlepropagations.Thistypeofusageofoptical-latticequantumsimulators
is rather natural for and highly demanded by theoretical researchers, for whom the
contents in Chap. 5 will be a simple textbook example.
Higashi-Osaka, Japan Prof. Ippei Danshita
May 2020
