Dynamical generation of artificial gauge fields in optical...
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Dynamical generation of artificial gauge fields
in optical lattices
André Eckardt [email protected]
Max-Planck-Institut für Physik komplexer Systeme Dresden
International School Anyon Physics of Ultracold Atomic Gases
Freie Universität Berlin September 27, 2013
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Plan of the lectures Introduction • Ultracold atoms in optical lattice potentials • Representation of magnetic fields in tight-binding lattices • Artificial magnetic fields for neutral atoms in optical lattices Quantum engineering in time 𝐻 𝑡 + 𝑇 = 𝐻 𝑡 ⟹ 𝐻eff • Quantum Floquet theory • Perturbative computation of 𝐻eff
Dynamical generation of magnetic fields in tight-binding lattices • General scheme • Application 1: Staggered-flux triangular lattice (kinetic frustration) • Application 2: Engineering the Harper Hamiltonian
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Plan of the lectures Introduction • Ultracold atoms in optical lattice potentials • Representation of magnetic fields in tight-binding lattices • Artificial magnetic fields for neutral atoms in optical lattices Quantum engineering in time 𝐻 𝑡 + 𝑇 = 𝐻 𝑡 ⟹ 𝐻eff • Quantum Floquet theory • Perturbative computation of 𝐻eff
Dynamical generation of magnetic fields in tight-binding lattices • General scheme • Application 1: Staggered-flux triangular lattice (kinetic frustration) • Application 2: Engineering the Harper Hamiltonian
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Ultracold atomic quantum gases
attractive field
atoms
vacuum chamber
laboratory (T ~ 300 K)
Trap atoms
atoms
red-detuned lasers Laser cooling
time
Evaporative cooling to quantum degeneracy:
𝑇~ nano Kelvin 𝑁~ 1 to 108 𝑁𝑉~ 1013 to 1015cm-3 (air: 1019cm-3, solids: 1022cm-3)
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Ultracold atomic quantum gases
attractive field
atoms
vacuum chamber
laboratory (T ~ 300 K)
Trap atoms
atoms
red-detuned lasers Laser cooling
time
Evaporative cooling to quantum degeneracy:
𝑇~ nano Kelvin 𝑁~ 1 to 108 𝑁𝑉~ 1013 to 1015cm-3 (air: 1019cm-3, solids: 1022cm-3)
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Ultracold atomic quantum gases
attractive field
atoms
vacuum chamber
laboratory (T ~ 300 K)
Trap atoms
atoms
red-detuned lasers Laser cooling
time
Evaporative cooling to quantum degeneracy:
𝑇~ nano Kelvin 𝑁~ 1 to 108 𝑁𝑉~ 1013 to 1015cm-3 (air: 1019cm-3, solids: 1022cm-3)
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Description
• clean & well isolated from environment • universal contact interactions
• taylorable and controllable, also during experiment
• additional “features” possible fermions, spin, dissipation, disorder, …, artificial magnetic fields, …
Spinless bosons:
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Optical Lattices standing light wave
clean periodic potential
Deep lattices
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Optical Lattices
Deep lattices
Ratio 𝑈/𝐽 tunable via laser power: from weak to strong coupling regime
Described by Hubbard models Jaksch et al., PRL (1998)
bosons
J
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Optical Lattices
Different lattice geometries / reduction to1D or 2D
Ratio 𝑈/𝐽 tunable via laser power: from weak to strong coupling regime
Described by Hubbard models Jaksch et al., PRL (1998)
bosons
J
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Cold-atom lattice systems • clean & tunable realizations of minimal many-body models • strong interactions possible • well isolated from environment • time-dependent parameter control • few-particle correlations directly measurable (single-site resolution) => quantum engineering of many-body systems • push boundaries of human control over quantum behavior • study exotic equilibrium physics • study coherent many-body quantum dynamics • …
today: • time-periodically driven optical lattices • how to effectively create artificial gauge fields for neutral atoms
J
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External fields in tight-binding lattices Vector potential represented by Peierls phases Scalar potential represented by on-site energies
Magnetic flux through a lattice plaquette P
2 1
4 3
𝑒𝑖Θ21
𝑒𝑖Θ32
𝑒𝑖Θ43
𝑒𝑖Θ14 Flux quantum Φ0 = 2𝜋
Invariant under gauge transformations
Constant vector potential:
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• Complete the toolbox for mimicking charged particles
• Reach Quantum Hall regime # magnetic flux quanta ~ # particles
• Intriguing interplay between lattice and gauge field – strong-field regime (fractal Hofstadter butterfly spectrum relevant)
# magnetic flux quanta ~ # lattice cells
– Chern/topological insulators
gauge-field changes on length scale of the lattice
→ Bloch bands with quantized (spin) Hall conductivity (like Landau level)
• Intriguing interplay with interactions
– Fractional Quantum Hall effect / Fractional Chern insulators
– Mimic quantum antiferromagnetism with hard-core bosons
Why artificial gauge fields in optical lattices?
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• Complete the toolbox for mimicking charged particles
• Reach Quantum Hall regime # magnetic flux quanta ~ # particles
• Intriguing interplay between lattice and gauge field – strong-field regime (fractal Hofstadter butterfly spectrum relevant)
# magnetic flux quanta ~ # lattice cells
– Chern/topological insulators
gauge-field changes on length scale of the lattice
→ Bloch bands with quantized (spin) Hall conductivity (like Landau level)
• Intriguing interplay with interactions
– Fractional Quantum Hall effect / Fractional Chern insulators
– Mimic quantum antiferromagnetism with hard-core bosons
Why artificial gauge fields in optical lattices?
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Using internal atomic structure State-dependent lattices + Laser-assisted tunneling • Jaksch & Zoller, NJP 2003 • Mueller, PRA 2004 • Gerbier & Dalibard, NJP 2010 Optical Flux lattice Lattice and gauge field created on same footing • Cooper PRL 2011 • Dalibard & Cooper, EPL 2011 • Cooper & Moessner, PRL 2012 • Juzeliūnas & Spielman, NJP 2012 • Dalibard & Cooper, PRL 2013
Proposals for non-abelian gauge fields • Osterloh et al. PRL 2005 • … more … Experiment: tunable1D gauge potential Jiménez-García et al PRL 2012 (Spielman)
How to create artificial gauge fields in optical lattices?
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Dynamically w/o internal structure Lattice shaking (kHz-regime) • EPL 89, 10010 (2010)
π-flux triangular lattice • PRL 108, 225304 (2012 )
tunable magnetic fields • PRL 109, 145301 (2012)
Chern/topological insulators, non-abelian gauge fields
Moving superlattice • Kolovsky EPL (2011)
(tuanble) magnetic fields
Stirring potentials • Lim, Morais Smith & Hemmerich PRL (2008)
tunable stagered square lattice • Kitagawa et al. PRB (2010)
topological insulatior on hexagonal lattice
Using internal atomic structure State-dependent lattices + Laser-assisted tunneling • Jaksch & Zoller, NJP 2003 • Mueller, PRA 2004 • Gerbier & Dalibard, NJP 2010 Optical Flux lattice Lattice and gauge field created on same footing • Cooper PRL 2011 • Dalibard & Cooper, EPL 2011 • Cooper & Moessner, PRL 2012 • Juzeliūnas & Spielman, NJP 2012 • Dalibard & Cooper, PRL 2013
Proposals for non-abelian gauge fields • Osterloh et al. PRL 2005 • … more … Experiment: tunable1D gauge potential Jiménez-García et al PRL 2012 (Spielman)
How to create artificial gauge fields in optical lattices?
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Dynamically w/o internal structure Lattice shaking (kHz-regime) • EPL 89, 10010 (2010)
π-flux triangular lattice • PRL 108, 225304 (2012 )
tunable magnetic fields • PRL 109, 145301 (2012)
Chern/topological insulators, non-abelian gauge fields
Moving superlattice • Kolovsky EPL (2011)
(tuanble) magnetic fields
Stirring potentials • Lim, Morais Smith & Hemmerich PRL (2008)
tunable stagered square lattice • Kitagawa et al. PRB (2010)
topological insulatior on hexagonal lattice
First experiments:
𝜋 𝜋 𝜋 𝜋
𝜋 𝜋 𝜋 𝜋 Science 333, 996 (2011)
How to create artificial gauge fields in optical lattices?
Nature Phys. (2013) doi:10.1038/nphys2750
Φ Φ
Φ Φ −Φ
−Φ −Φ
−Φ
Aidelsburger et al. PRL (2011)
Aidelsburger et al. arXiv:1308.0321 Miyake eta l. arXiv:1308.1431
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Plan of the lectures Introduction • Ultracold atoms in optical lattice potentials • Representation of magnetic fields in tight-binding lattices • Artificial magnetic fields for neutral atoms in optical lattices Quantum engineering in time 𝐻 𝑡 + 𝑇 = 𝐻 𝑡 ⟹ 𝐻eff • Quantum Floquet theory • Perturbative computation of 𝐻eff
Dynamical generation of magnetic fields in tight-binding lattices • General scheme • Application 1: Staggered-flux triangular lattice (kinetic frustration) • Application 2: Engineering the Harper Hamiltonian
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Time-periodic Hamiltonian (Floquet system)
Useful? Yes! If has simple form (at least approximatly)
Effective time-independent Hamiltonian for time-evolution over one period:
Quantum engineering in time: Engineer a time-periodic many-body system that realizes an effective time-independent Hamiltonian of interest!
In a nutshell
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possesses solutions equivalently
Floquet states Schrödinger equation with time-periodic Hamiltonian
Floquet state Quasienergy
Floquet mode
Floquet states form complete orthonormal basis at every time 𝑡
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Floquet states Proof:
Time evolution operator:
Monodromy operator:
• Eigenstates form complete orthonormal basis (from unitarity)
• Eigenvalues are phase factors (from unitarity)
• Eigevalues are independent of 𝑡 (from )
Eigenstates are Floquet states
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Time evolution generated by time-periodic Hamiltonian
• If prepared in Floquet state: purely periodic
• If prepared in superposition of Floquet states: stroboscopic time-evolution determined by quasienergies 𝜀𝛼
How to compute Floquet states and quasienergy practically? numerically ? analytic approximations?
Time evolution on short times within one period (micromotion)
Long-time behavior
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Eigenvalue problem of monodromy operator
use together with
Usefull for numerical computation of small systems: • Compute by integrating the time-evolution
for a complete set of basis states
• Diagonalize fully
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Quasienergy eigenvalue problem [Sambe PRA (1973)]
Ambiguity in definition of Floquet modes |𝑢𝛼(𝑡)⟩ (here 𝜔 = 2𝜋𝑇
)
Hermitian quasienergy eigenvalue problem (time plays role of a coordinate)
Quasienergy operator
− drastically enlarged Hilbert space
+ Stationary perturbation theory applicable
+ Adiabatic principle works
+ Intuitive Framework for resonance effects
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The Floquet Picture Arbitrary time-dependent Hamiltonian
Two-times formalism
Consider generalized Schrödinger equation in extended Hilbert space:
Project back to original state space:
Use tools and intuition of non-driven systems Stationary perturbation theory for eigenvalue problem of
Adiabatic principle for parameter variation
with
e.g.:
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Perturbation theory for effective Hamiltonian Quasienegy eigenvalue problem
Appropriately chosen basis
Strategy for choosing Integrate out large terms ~ℏ𝜔
If
neglect off-diagonal blocks
⇒ effective Hamiltonian
(1st order degenerate perturbation theory,
systematic corrections from higher orders)
𝑚 plays role of a ``photon‘‘ number
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Plan of the lectures Introduction • Ultracold atoms in optical lattice potentials • Representation of magnetic fields in tight-binding lattices • Artificial magnetic fields for neutral atoms in optical lattices Quantum engineering in time 𝐻 𝑡 + 𝑇 = 𝐻 𝑡 ⟹ 𝐻eff • Quantum Floquet theory • Perturbative computation of 𝐻eff
Dynamical generation of magnetic fields in tight-binding lattices • General scheme • Application 1: Staggered-flux triangular lattice (kinetic frustration) • Application 2: Engineering the Harper Hamiltonian
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Basic scheme for generation of gauge fields
Hubbard Hamiltonian with periodic driving
tunneling periodic driving
possible static tilt
weak trap, interactions,
Unitary transformation (interaction picture / change of gauge)
Time average over one period
Effective tunneling matrix elements (can be complex)
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Basic scheme for generation of gauge fields
Plaquette fluxes Φ𝑃 = 0,𝜋 (time-revresal symmetry not broken)
if global reflection symmetry
These symmtries also prevent ratchet-type rectification Flach et al. PRL 84, 2358 (2000), Denisov et al. PRA 75, 063424 (2007).
if the and local reflection symmetry or shift antisymmtry
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Basic scheme for generation of gauge fields
Case 1: AC-modified tunneling (no off-sets )
Case 2: AC-induced tunneling (strong off-sets )
Plaquette fluxes Φ𝑃 ≠ 0,𝜋 requires to break
Plaquette fluxes Φ𝑃 ≠ 0,𝜋 requires to break
Easier to break, e.g. a moving Superlattice is enough
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AC-modified tunneling via lattice shaking inertial force
Break symmetry
Square plaquettes remain trivial Φ𝑃 = 𝜃 + 𝜃′ − 𝜃 − 𝜃′ = 0
𝜃
𝜃
𝜃𝜃 𝜃𝜃
Triangular plaquette flux tunable Φ𝑃 = 𝜃 − 2𝜃′ ≠ 0 𝜃𝜃 𝜃𝜃
𝜃
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AC-induced tunneling in tilted lattice via moving superlattice
(Kolovsky proposal & Bloch/Ketterle experiments)
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J
Dynamically induced quantum phase transition
superfluid particles delocalized, gapless phonon excitations
Mott-insulator particles localized at sites, gapped particle-hole excitations
Gre
iner
et a
l., N
atu
re (2
002)
MPA Fisher et al., PRB (1989), for cold atoms: Jaksch et al., PRL (1998)
bosonic ground state:
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J
Dynamically induced quantum phase transition
superfluid particles delocalized, gapless phonon excitations
Mott-insulator particles localized at sites, gapped particle-hole excitations
Gre
iner
et a
l., N
atu
re (2
002)
MPA Fisher et al., PRB (1989), for cold atoms: Jaksch et al., PRL (1998)
bosonic ground state:
experiment: Zenesini et al., PRL (2009) proposal: Eckardt et al., PRL (2005)
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Perturbation theory for effective Hamiltonian Quasienegy eigenvalue problem
Appropriately chosen basis
Strategy for choosing Integrate out large terms ~ℏ𝜔
If
neglect off-diagonal blocks
⇒ effective Hamiltonian
(1st order degenerate perturbation theory,
systematic corrections from higher orders)
𝑚 plays role of a ``photon‘‘ number
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Eckardt & Holthaus, PRL 2008
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Dynamically induced frustration
in a triangular lattice
Joint work with experimentalists from Sengstock group in Hamburg
Struck et al., Science 2011 Eckardt et al. EPL 2010
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Shaken triangular lattice Elliptically shaken triangular lattice
Frustrated kinetics for
J J‘ 𝜋 𝜋 𝜋 𝜋
𝜋 𝜋 𝜋 𝜋
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Limit of weak interactions Condensate with wave function:
resemble classical rotors with antiferromagnetic coupling
Experiment Reciprocal lattice
Dispersion relation
1 --
0 --
2 --
Direct lattice
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Spontaneous breakingof time-reversal symmetry
Circular plaquette currents
Condensate with wave function:
resemble classical rotors with antiferromagnetic coupling
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Corrections for intermediate interaction
condensate fraction = 0.75
Spiral staggered
“Order-by-disorder-type effect”
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Strong interaction Hard-core boson limit System resembles frustrated quantum antiferromagnet
Ground state difficult to predict
Two simple Ansaetze give the same energy per spin -(3/8) J
Classical Neel order
Cover of singlets (exponentially degenerate!) => Valence bond solid or Spin liquid (gapped or critical)
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Strong interaction Hard-core boson limit System resembles frustrated quantum antiferromagnet
Novel type of quantum spin simulator
• built on easy-to-cool bosonic motional („charge“) degrees of freedom
• large coupling of the order of boson tunneling (no superexchange)
• different adiabatic preparation schemes (tunable frustration & „quantumness“)
• can host quantum disordered spin-liquid-like phases
• generalizable to further lattice geometries, e.g. Kagome (Berkeley group)
• easy to implement experimentally
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Strong interaction Hard-core boson limit System resembles frustrated quantum antiferromagnet
Conjectured phase diagram at half filling:
Schmied et al. NJP 2008: PEPS and exact diagonalization
J‘/J ~1.4 |
~1.2 |
~0.6 |
~0.4 |
0 |
Staggered Neel order
Spiral Neel order
algebraic
gapped spin liquid
gapped spin liquid
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Conclusions Lattice shaking is a low-demanding method for the creation of artificial gauge fields (both abelian and non-abelian) for neutral atoms. Opens novel routes for engineering many-body physics in optical lattices
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Thanks to collaborators of the presented work
Barcelona (Theory): Maciej Lewenstein Philipp Hauke (now in Innsbruck) Olivier Tieleman (now in Dresden) Alessio Celi
Hamburg (Experiment) Sengstock group
Rodolphe LeTargat (now Paris), Christoph Oelschlaeger, Klaus Sengstock, Juliette Simonet, , Parvis Soltan-Panahi (now Bosch GmbH), Julian Struck, Malte Weinberg, Patrick Windpassinger (now Mainz)