Hanns-Christoph NägerlInstitut für Experimentalphysik, Universität Innsbruck
Dresden, July 5th 2019
Quantum dynamics in strongly correlated one-dimensional Bose gases
2012-2016
2019-2023Wittgenstein
Three quantum gas projects in my group in Innsbruck
CsIIICs and Cs2 in optical lattices:
1D physicsRb-CsRb-Cs mixtures
and bosonic RbCs dipolar molecules
K-CsK-Cs mixtures,
quantum gas microscopy,fermionic dipoles, topological states
Innsbruck Cesium-Team
FlorianMeinert
Manfred Mark
EmilKirilov
KatharinaLauber Philipp
Weinmann
Michael Gröbner
Andrew Daley
HCN
Theory support
The Team & Collaborators…
Michael KnapEugene Demler Mikhail Zvonarev
Jean-Sébastien Caux
Dirk-Sören Lühmann
Milosz Panfil
Ole Jürgensen
Team Members now
Milena Horvath,
PhD-Student
AnamikaNair,
Intern
BeatrixMair
(Rb-Cs)ManueleLandini,
Senior Scientist
BodhaSantra,
Postdoc
Spring 2019
Moore’s law
Gordon Moore
Oct. 2018: Samsung announced that it has started commercialproduction of 7-nm scale technology (~13 Si atoms)
Source: Intel
... approaching the ultimate limitof single-atom wires
B. Weber et al., Science 335, 64 (2012)
Electronic motion at the nano-scale
STM image of an atomic wire in Si,4 atoms wide and 1 atom tall
Quantum phenomenain 1D single-atom wires:
• many-body quantumdynamics and transport
• role of particle interactionsand correlations
• specifically in 1D-systems
• time-resolved observations
ultracold atomsconfined to 1D
„quantum tubes“
50 nm
Strongly interacting atoms in 1D
force F
Many-body dynamicson a lattice
Bose-Hubbard Hamiltonian
Nearest-neighbortunneling
on-siteinteraction
M. Greiner et al., Nature 415, 39 (2002)
Strongly interacting atoms in 1D
force F
Many-body dynamicson a lattice
Bose-Hubbard Hamiltonian
Nearest-neighbortunneling
on-siteinteraction
M. Greiner et al., Nature 415, 39 (2002)
scattering length in cesium
strongly interacting 1D quantum fluid
Beyond static latticepotentials
„quantum wire“ withtunable interactions
E. Haller et al., Science 325, 1224 (2009)
Strongly interacting atoms in 1D
force F
Many-body dynamicson a lattice
Bose-Hubbard Hamiltonian
Nearest-neighbortunneling
on-siteinteraction
M. Greiner et al., Nature 415, 39 (2002)
• crystal-like properties
• collective excitations
• low-energy sector:
Luttinger liquidphonons!
Beyond static latticepotentials
„quantum wire“ withtunable interactions
E. Haller et al., Science 325, 1224 (2009)
Strongly interacting atoms in 1D
force F
Many-body dynamicson a lattice
Bose-Hubbard Hamiltonian
Nearest-neighbortunneling
on-siteinteraction
M. Greiner et al., Nature 415, 39 (2002)
Beyond static latticepotentials
„quantum wire“ withtunable interactions
E. Haller et al., Science 325, 1224 (2009)
• crystal-like properties
• collective excitations
• low-energy sector:
Luttinger liquidphonons!
Example: 1D Mott insulator
J. Simon et al., Nature 472, 307 (2011) S. Sachdev et al., Phys. Rev. B 66, 075128 (2002)
J«UMott insulator
long-range correlatedtunnel transport
correlated tunneling motionin strong fields
E=U/2 E=U/3E=U
E
doublons produced after 200 ms
F. Meinert, M. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner,A.J. Daley, and H.-C. Nägerl, Science 344, 1259 (2014)
Resonant long-range transport
Long-range tunneling
E=UE=U/2
E=U/3
V=10ER
F. Meinert, M. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner,A.J. Daley, and H.-C. Nägerl, Science 344, 1259 (2014)
Resonant long-range transport
E=U/2
J J
virtual state
analogue at the nano-scale
electron transport in quantum dot arrays
F. Braakman et al., Nat. Nanotech. 8, 432 (2013)
Long-range tunneling
F. Meinert, M. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner,A.J. Daley, and H.-C. Nägerl, Science 344, 1259 (2014)
Resonant long-range transport
long-distance atom transport
without occupying intermediatelattice sites
analogues at the nano-scale
electron transport in quantum dot arrays
F. Braakman et al., Nat. Nanotech. 8, 432 (2013)
E=U/2
E=U/3
E=U/4
E=U/5
V=7ER
Long-range tunneling
E=1.038 kHz
E=0.973 kHz
E=0.865 kHz
F. Meinert, M. Mark, E. Kirilov, K. Lauber, P. Weinmann, A.J. Daley,and H.-C. Nägerl, Phys. Rev. Lett. 111, 053003 (2013)
U=1.019(20) kHz
Time-resolved tunneling
E≈U
Coherent many-body tunneling dynamics
Coherent tunnel coupling
V=10ER
Dynamics driven by higher-order tunneling
Second- and third-order tunneling
E=U/37 ER, 253 a09 ER, 253 a09 ER, 175 a0
E=U/2
Growth rate 1/τDepends onJ and U.
10 ER, 253 a012 ER, 253 a010 ER, 400 a0
F. Meinert et al., Science 344, 1259 (2014)
Bloch oscillations in a latticeen
ergy
(ar
b. u
.)
quasi-momentum q (hkl)
single particle band structure
coherentBloch oscillations
J»USuperfluid
F. Meinert et al., Phys. Rev. Lett. 112, 193003 (2014)
BOs in momentum space
F
Tilting a 1D superfluid
non-interacting (U=0)
Bloch oscillations
th= 0 TB th= 7 TB th= 14 TB
Vz=7ER (J=52.3 Hz)
TB≈0.57 ms F. Meinert, M.J. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner,and H.-C. Nägerl, Phys. Rev. Lett. 112, 193003 (2014)
Tilting a 1D superfluid
th= 0 TB th= 7 TB th= 14 TB
Collapse and revivals
Vz=7ER (J=52.3 Hz)
TB≈0.57 ms
weakly-interacting (U=110 Hz)
Tilting a 1D superfluid
Period of the phase revival
frev = U/hDirect measure of U
in the superfluid regime
revi
val f
requ
ency
fre
v(H
z)
Blo
ch fre
quen
cy f
B(k
Hz)
Bloch oscillations in a lattice
dampedBloch oscillations
J≈U≈ESuperfluid
Superfluid sample in true 1D
Irreversible damping!
E = 266 Hz (squares)
E = 855 Hz (triangles)
Vz = 4 ER (i.e. J = 114.2 Hz)U = 106 Hz
Superfluid sample in true 1D
as = 4.5 a0 (squares), as = 20.8 a0 (triangles), as = 83.4a0 (circles) at E = 346(10)Hz and Vz = 4 ER (i.e. J = 114.2 Hz)
Irreversible damping as a fct of interaction strength
Superfluid sample in true 1D: Energy spectrum
E ≈ J ≈ U E >> J ≈ U
Theory:exact diagonali-zationfor 5 particleson 5 sites
Superfluid sample in true 1D: Damping rate
E = 346 Hz and Vz = 4 ER (J = 114.2 Hz) F. Meinert et al., PRL 2014
Strongly interacting atoms in 1D
F
Many-body dynamicson a lattice
Bose-Hubbard Hamiltonian
Nearest-neighbortunneling
on-siteinteraction
M. Greiner et al., Nature 415, 39 (2002)
• crystal-like properties
• collective excitations
• low-energy sector:
Luttinger liquidphonons!
Beyond static latticepotentials
„quantum wire“ withtuneable interactions
E. Haller et al., Science 325, 1224 (2009)
Can there be Bloch oscillations without a
lattice?
quasi-momentum q (hkl)
ener
gy (
arb.
u.)
Bloch oscillations without a lattice
Theory proposal
Bloch oscillations without a lattice
short-range crystal-likeproperties via strong repulsion
E. Lieb and W. Liniger, Phys. Rev. 130, 1616 (1963)
D. M. Gangardt et al., Phys. Rev. Lett. 102, 070402 (2009)M. Knap et al., Phys. Rev. Lett. 112, 015302 (2014)
ImpurityBloch oscillations
Tonks-Girardeau gas (TG)
low-energyspectral edge
Impurity
Model: E. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963)
bosons in uniform 1D systemrepulsive contact potential
Hamilton operator:
γ − parameter
Tonks-Girardeau gas (TG)( non-interacting fermions )( hard spheres )
kinetic energy interaction energy
Ideal gas(non-interacting bosons)
Lieb - Liniger model: Fermionization
c - constant γ - interaction strength
sketch: wave functions for two particles in harmonic trap
bosons
repulsiveinteraction
attractiveinteraction
fermions
Bose-Fermi mapping
Bose-Fermi mapping: bosons and fermions in 1D show similar density distributions
Bose-Fermi mapping
non-interactingstrongly attractive
fermionsbosons
sketch: wave functions for two particles in harmonic trap
Bose-Fermi mapping: bosons and fermions in 1D show similar density distributions
Bose-Fermi mapping
strongly repulsive
non-interacting
fermionsbosons
sketch: wave functions for two particles in harmonic trap
Bose-Fermi mapping: bosons and fermions in 1D show similar density distributions
Consequence for the Mott-insulator transition
Mott – insulator phase transition“metal - insulator transition”
deep lattice, tight-binding approximationconnects ground states of the
Bose-Hubbard modelsuperfluid Mott insulator
Mott-Hubbard transition
lattice
Pinning transition
add shallow lattice (perturbation) connects ground states of the
sine-Gordon modelTonks gas Mott insulator
My naïve intuition
pair correlation function
position0
Impurity performs Bloch oscillationson the g(2) -function!!!
Tuning interactions
Interaction control for impurity
Notation: for the bath-bath interaction
for the impurity-bath interactionscatterering length data by P. Julienne
host gas: F=3, mF=3
impurity: F=3, mF=2
Impurity transport: Experiment
Impurity momentum distribution n(k) for no interactions: Free fall
dashed line: free fall with g/3
Impurity momentum distribution: Experiment vs. theory
experimental resolution
Fermi k-vector, Fermi time
F. Meinert et al., Science 356, 945 (2017)
Impurity momentum distribution: Experiment vs. theory
experimental resolution
Fermi k-vector, Fermi time
F. Meinert et al., Science 356, 945 (2017)
Impurity transport: Experiment vs. theoryMean impurity momentum green solid line: theory, no free parameters
F. Meinert et al., Science 356, 945 (2017)
Impurity transport: Comparision with theory
Bloch frequency drift momentum
Force in dimensionless units
F. Meinert et al., Science 356, 945 (2017)
Impurity transport: Comparision with theory
Bloch frequency drift momentum
Force in dimensionless units
F. Meinert et al., Science 356, 945 (2017)
Impurity transport: Can one do better?
Yes, use a weaker force!
Impurity transport: Many more questions
Where does the energy go? (origin of the damping)
What happens for even stronger interactions?
What happens for attractive interactions?(either within the bath and/or between impurity and bath)
What happens for more than one impurity per tube?
...
What happens for mass-unbalanced systems?Can one use the impurity as a probe for phasetransitions? (e.g. „pinning“ transition)
see E. Haller et al., Nature 466, 597 (2010)
What happens for uniformly moving impurities?see proposal by C. Mathey, M. Zvonarev, and E. Demler, Nature Physics 8, 881 (2012)
see E. Haller et al., Science 325, 1224 (2009)
Impurity transport: “quantum flutter” proposal
free Fermi gas
Impurity transport: “quantum flutter” proposal
Impurity transport: “quantum flutter” proposal
Thank you!
One-dimensional systems are special!
Kinematics: Scattering in D=1 is special
Scattering in dimensions
?Scattering in dimension
Kinematics determines out-states in D=1 uniquely
Single-particledispersion:
Single particle-holepair spectrum:
Fermi momentum
Density
Particle branch
Hole branchhole
particle
Single particle-hole pair: some excitations are kinematically forbidden
Example: The D=1 non-interacting Fermi gas
πn
n
Example: Excitations in the D=1 Bose gas
Dynamic structure factor
free fermionsWealy-interactingbosons
J.-S. Caux and P. Calabrese,Phys. Rev. A 74, 031605(R) (2006)
increase interactions
Here: Quantum wires of bosons
Array of 1D tubes
Quantum wires of bosons
Lieb-Liniger model (1D gas of bosons, repulsive contact potential, T=0)
Lieb-Liniger γ−parameter
Tonks-Girardeau gas (TG)(fermionization)
kinetic term interaction term
ideal gas(non-interacting bosons)
c - constant γ - interaction strength
gas in 1D geometry
E. Haller et al., Science 325, 1224 (2009)
Quantum wires of bosons: Important parameters
Lieb-Liningerinteraction parameter Fermi k-vector
Ways to characterize the strongly-interacting D=1 Bose gas
How to characterize the strongly-interacting 1D gas?
Measure local correlations
Example: The D=1 Bose gas, local g(3) correlations
NIST group2004
Example: The D=1 Bose gas, local g(2) correlations
PennStategroup2004
Example: The D=1 Bose gas, local g(3) correlations
Innsbruckgroup2011
Other ways to characterize the D=1 Bose gas
Measure global properties
The strongly-interacting D=1 Bose gas
PennStategroup2004
saturation of samplelength as interactionsare increased
Characterization of the Tonks-Girardeau gas
Regimes
Noninteracting (thermal) 41D mean field 3Fermionized bosons (TG) 4... ...
Probe: Measure collective oscillation frequencies
Theory: C. Menotti and S. Stringari, PRA 66, 043610 (2002)
compression dipole
Characterization of the crossover to the Tonks-Girardeau gas...
E. Haller et al.,Science 325, 1224 (2009)
... and to the super Tonks-Girardeau gas
E. Haller et al.,Science 325, 1224 (2009)
A2 ~ 1/γ2 interaction parameter
hard rods
Monte Carlo
red data:
blue data:
Another method: Measurement of the excitation spectrum
Our new method: Bragg spectroscopy
Direct measurement of the dynamic structure factor
Elementary excitations of the 1D Bose gasE. Lieb and W. Liniger, Phys. Rev. 130, 1616 (1963)Dynamic structure factor
free fermionsWeakly-interacting bosons
J.-S. Caux and P. Calabrese,Phys. Rev. A 74, 031605(R) (2006)
Interesting regime
= 1 to 10
finite temperature
and
Probing excitations via Bragg spectroscopyFlorence group: N. Fabbri et al., Phys. Rev. A 91, 043617 (2015)
Bragg spectroscopy in 1D
1D Bose gas with tuneable interactions
repulsive
attractive
confinement + Cs Feshbach structure
Time of flight---------
mapping tomomentum space
measure energy transfer
confinement-inducedresonance (CIR)
allows us to tune
E. Haller et al.,Science 325, 1224 (2009)
Bragg spectroscopy in 1D:Theory expectation
zero temperature expectation insets: thick solid curves arewith inhomomgenous density
Bragg spectroscopy in 1D
fixed momentum transfer
Bragg spectroscopy in 1D
Excitation spectrum at finite temperatures
Theory: Y. Cherny and J. Brand,Phys. Rev. A 73, 023612(R) (2006)
Comparison with theory tells us that we see a collective excitation!
Details: F. Meinert et al., Phys. Rev. Lett. 115, 085301 (2015)
¼ 11 ¼ 22 ¼ 45
M. Panfil and J.-S. Caux,Phys. Rev. A 89, 033605 (2014)
Theory prediction: Momentum space images (for comparatively weak interactions)
Interaction control for impurity: Theory
Force in dimensionless units
Interaction control for impurity: Theory
Calc. based onmatrixproductstates
Interaction control for impurity: Theory
Calc. based onmatrixproductstates
PublicationsQuantum Quench in an Atomic One-Dimensional Ising ChainF. Meinert, M. Mark, E. Kirilov, K. Lauber, P. Weinmann, A.J. Daley, and H.-C. Nägerl, Phys. Rev. Lett. 111, 053003 (2013)
F. Meinert, M.J. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner, and H.-C. Nägerl, Phys. Rev. Lett. 112, 193003 (2014)
Interaction-Induced Quantum Phase Revivals and Evidence for the Transition to the Quantum Chaotic Regime in 1D Atomic Bloch Oscillations
F. Meinert, M. Panfil, M. J. Mark, K. Lauber, J.-S. Caux, and H.-C. Nägerl, Phys. Rev. Lett. 115, 085301 (2015)
Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling
F. Meinert, M. J. Mark, K. Lauber, A. J. Daley, and H.-C. Nägerl, Phys. Rev. Lett. 116, 205301 (2016)
Floquet Engineering of Correlated Tunneling in the Bose-Hubbard Model with Ultracold Atoms
O. Jürgensen, F. Meinert, M. Mark, H.-C. Nägerl, and D.-S. Lühmann,Phys. Rev. Lett. 113, 193003 (2014)
Observation of Density-Induced Tunneling
F. Meinert, M. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner, A.J. Daley, and H.-C. Nägerl, Science 344, 1259 (2014)
Observation of many-body dynamics in long-range tunneling after a quantum quench
F. Meinert, M. Knap, E. Kirilov, K. Jag-Lauber, M. B. Zvonarev, E. Demler, and H.-C. Nägerl, Science (2017)
Bloch oscillations in the absence of a lattice
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