Simulating dynamical abelian and non-abelian gauge theories
with ultracold atoms
Benni Reznik Tel-Aviv University
1
Joint work with
Erez Zohar, Tel-Aviv University J. Ignacio Cirac, MPQ
Trento, July 5, 2013
Quantum simulations
β’ Condensed Matter Models ( high TC?) Artificial potentials (Topological insulators, Hall effect) β’ Gravity: Black hole Hawking radiation, quantum effects of cosmological expansion. β’ QFT: Scalar and fermion relativistic fields,
β’ Condensed Matter Models ( high TC?) Artificial potentials (Topological insulators, Hall effect) β’ Gravity: Black hole Hawking radiation, quantum effects of cosmological expansion. β’ QFT: Scalar and fermion relativistic fields,
β’ .
β’ High Energy physics (HEP) β’ Quantum Chromodynamics (QCD) effects? β’ Confinement of Quarks?
Quantum simulations
Simulating systems
Using one controllable quantum system in order to simulate another system,
for example by:
β’ BECs β’ Atoms in optical lattices β’ Rydberg Atoms β’ Trapped Ions β’ Superconducting devices β’ β¦
4
Simulating HEP Physics
β’ Experimentally more challenging compared with cond. mat. simulations.
β’ Could be used to explore otherwise experimentally and computationally inaccessible effects, and directly unobservable phenomena at short scales.
β’ There are different proposals with different levels of difficulty.
Starting with relatively simple models, that can
be developed to study complex challenging problems -> Standard model?
Kapit & Mueller: continuum, Abelian (QED), w. atoms. Analog.
P. Zoller group: lattice Abelian/Non-Abelian: w. atoms, ions, SQC. Analog.
M. Lewenstein group: lattice Abelian/Non-Abelian,Rydberg atoms. Digital
Talk outline
I) HEP and Lattice gauge theory β an overview Fundamental physical requirements. Structure of HEP models. Abelian vs. Non-abelian gauge field physics. The structure of lattice gauge theory (LGT).
II) Simulating LGT with ultracold atoms . Basic ideas: how to realize Links, plaquettes.
Examples: simulating Quark confinement, vacuum fluctuations, Wilson loops.
III) QCD: Non-abelian Yang-Mills gauge theory.
β’ Outlook and Summary
Requirements: HEP models
I) fields:
Fermion fields := Matter
Bosonic gauge fields := Interaction mediators
II) local gauge invariance => βchargeβ conservation (exact, or low energy, effective )
III) Relativistic invariance => causal structure ( In the continuum limit.)
=> dynamics and interactions
Structure of HEP models
β’ Matter particles:= fermions (+Higgs) (Charges: electric, color , flavor. Spin. Mass)
β’ Gauge fields:= mediate the Interactions
Electromagnetic: massless photon, (1), U(1)
Weak interaction : massive Z, Wβs , (3), SU(2)
Strong interaction : massless Gluons , (8), SU(3)
Structure of HEP models
The abelian/non-abelian local symmetries much determines the Gauge field physical properties and dynamics.
It also much constrains the allowed possible interactions between Gauge fields and Matter-field.
U(1) => linear Maxwell theory
SU(N) => non-linear Yang-Mills theory
πΌππΈπ· βͺ 1, πππΈπ· π β1
π
We (ordinarily) donβt need second quantization to understand
the structure of atoms: πππ2 β« πΈπ π¦πππππ β πΌππΈπ·
2 πππ2
Abelian: QED
πΌππΈπ· βͺ 1, πππΈπ· π β1
π
We (ordinarily) donβt need second quantization to understand
the structure of atoms. πππ2 β« πΈπ π¦πππππ β πΌππΈπ·
2 πππ2
Also in higher energies (scattering, fine structure corrections)
Perturbation theory ( Feynman diagrams) works well.
Abelian: QED
Abelian: QED
e.g. , the anomalous electron magnetic moment:
β¦+
+ +β¦
( 891) Feynman vertex diagrams)
(g-2)/2= β¦.
β¦+
12672 self energy diagrams
(g-2)/2= 1 159 652 180.73 (0.28) Γ 10β12 g β 2 measurement by the Harvard Group using a Penning trap
T. Aoyama et. al. Prog. Theor. Exp. Phys. 2012, 01A107
Non-abelian: QCD
πΌππΆπ· > 1 , πππΆπ· π β π
non-perturbative confinement effect!
=> structure of Hadrons: quark pairs form Mesons , quark triplets form Baryons. Color Electric flux-tubes: βa non-abelian Meissner effectβ.
Non-abelian: QCD
πΌππΆπ· > 1 , πππΆπ· π β π
non-perturbative confinement effect!
=> structure of Hadrons: quark pairs form Mesons , quark triplets form Baryons. Color Electric flux-tubes: βa non-abelian Meissner effectβ.
Gauge fields
Abelian Fields Maxwell theory
Non-Abelian fields Yang-Mills theory
Massless Massless
Long-range forces Confinement
Chargeless Carry charge
Linear dynamics Self interacting & NL
Lattice gauge theory β’ The standard avenue: ππππ‘π‘πππ β computed using
Monte Carlo βsamplingβ (Other methods exist: non-abelian bosonization (1+1); instanton semi-classical summation, large N perturbation, variational methodsβ¦ )
------------------
β¦ but:
β’ Correlations but not time dependence
β’ Limited applicability with too many quarks .
Computationally hard βsign problemβ.
(e.g. color superconductivity, Quark-Gluon plasma. )
β’ Quark confinement with dynamic matter never proven.
First step:
β’ Confinement in Abelian lattice models!
β’ Toy models with βQCD-likeβ properties that capture the essential physics mechanism
of confinement.
Confinement in abelian compact QED lattice models
β’ 1+1D: βSchwingerβs modelβ manifests confinement. (analytic and lattice results available).
β’ 2+1D: confinement for all values of coupling constant, a non-perturbative mechanism (Polyakov).
(For T > 0: there is a phase transition also in 2+1 D.)
β’ 3+1D: phase transition between strong coupling
confinement phase, and weak coupling coulomb phase.
β’ Z_N discrete gauge symmetry: in d+1, d>1. Electric and magnetic confinement & deconfinement
QED in 1+1 : Schwingerβs model
β’ No magnetic fields: EM has no dynamics of its own. Non trivial dynamics obtained by coupling to dynamical charge sources.
β’ Schwinger: π+πβ form bound states. (analytic and lattice results available.)
β’ Non-abelian extension: in 1+1: QπΆπ·2 version, not completely solved. Only in the large-N limit.
1+1, U(1) gauge theory
π» = ππππβ ππ
π
+ πΌπ ππβ ππ+1 +π». π.
Start with a hopping fermionic Hamiltonian, in 1 spatial direction
This Hamiltonian is invariant to global gauge transformations,
ππβΆ πβπΞππ ; ππ
β βΆ ππΞππβ
1+1, U(1) gauge theory
π» = ππππβ ππ
π
+ πΌπ ππβ ππππ+1 +π». π.
Promote the gauge transformation to be local:
Then, in order to make the Hamiltonian gauge invariant, add unitary operators, ππ,
ππβΆ πβπΞπππ ; ππ
β βΆ ππΞπππβ
ππ = ππππ ; ππβΆ ππ+ Ξπ+1 - Ξπ
(analogous to π΄ β π΄ + β Ξ )
Dynamic 1+1 U(1) gauge theory
Add dynamics to the gauge field:
Where πΏπ is the angular momentum operator
conjugate to ππ , representing the (integer) electric field.
π»πΈ =π2
2 πΏπ,π§
2
π
d+1 (d>1) : plaquette terms
In more spatial dimensions, we can add gauge-invariant interactions, plaquette terms:
In the continuum limit, this corresponds to π» Γ π¨ 2 - magnetic energy term.
β 1π2 cos ππ,π
1 + ππ+1,π2 β ππ,π+1
1 β ππ,π2
π,π
Figure from ref [6]
(m,n)
II.) Simulating LGT with ultracold atoms . Basic ideas: how to realize Links, plaquettes.
Some examples: simulating Quark confinement, vacuum fluctuations, Wilson loops.
Gauge fields: π΄ β π , πΈ β πΏ
Matter fields: π, πβ
Atomic analog simulator
Figure from ref [6]
General idea of our work
β’ Links: realized by atomic scattering : gauge invariance is
fundamental
β’ Plaquette: realized from gauge invariant building blocks, and Virtual loop contributions of ancillary fermions.
Figures from ref [6]
Realization of Links
Figure from ref [6]
FL FR
B1,B2
F
Fermion
FL FR
B1,B2
πΏ β πΏ β 1
F
Fermion
FL FR
B1,B2
F
Fermion
FL FR
B1,B2
πΏ β πΏ + 1
F
Fermion
Figure from ref [6]
Fermionic atom Bosonic atoms
Angular Momentum conservation Local gauge invariance
FL FR
B1,B2
ππΏβ π1β π2ππ + ππ
β π2β π1ππΏ
mF (FL)
mF (FR)
mF (B1)
mF (B1)
Angular Momentum conservation Local gauge invariance
FL FR
B1,B2
ππΏβ π1β π2ππ + ππ
β π2β π1ππΏ
mF (FL)
mF (FR)
mF (B1)
mF (B1)
Angular Momentum conservation Local gauge invariance
FL FR
B1,B2
ππΏβ π1β π2ππ + ππ
β π2β π1ππΏ
mF (FL)
mF (FR)
mF (B1)
mF (B1)
Gauge U(1) bosons: Schwinger algebra
πΏ+ = π1β π2 ; πΏβ = π2
β π1
πΏπ§ =1
2π1 β π2 ;π =
1
2π1 +π2
Gauge bosons: Schwinger algebra
πΏ+ = π1β π2 ; πΏβ = π2
β π1
πΏπ§ =1
2π1 β π2 ; π =
1
2π1 +π2
and thus what we have is
ππΏβ πΏ+ππ + ππ
β πΏβππΏ
ππΏβ π1β π2ππ + ππ
β π2β π1ππΏ
Gauge bosons: Schwinger algebra
ππΏβ πΏ+ππ + ππ
β πΏβππΏ
For large π , π βͺ π
πΏ+ = π1β π2~π
π π1βπ2 β‘ πππ = π
ππΏβ πππ + ππ
β πβ ππΏ
Qualitatively similar results can be obtained with just two bosons on the link.
Electric dynamic term
πΈ2 = πΏπ§2 =1
4π1 βπ2
2
=1
4π1β π12+1
4π2β π22β1
2π1β π1π2
β π2
π»πΈ =π2
2 πΏππ§
2
π
πΈ = ( π1 β π2)/2 πππππ’πππ‘π π‘π π = π1 β π2
Finally: cQED; U(1) in 1+1
π» = π β1 πππβ ππ
π
+ πΌ ππβ ππππ+1 +π». π.
+π2
2 πΏππ§
2
π
B-B Scattering: EM kinetic energy
F-B scattering: link interaction
The Schwinger model:
(βcβ=βcomactβ)
Mass and Charge
β’ Even n (particles):
π = π = πβ π 0 atoms: zero mass, zero charge
1 atom: M, Q=1
β’ Odd n (anti-particles):
π = π β 1 1 atom : zero mass, zero charge (βDirac seaβ)
0 atoms: mass M (relative to βM), charge Q=-1
note: mass is measured relative to βM
Quark confinement N =0 1 0 1
E= Lz = 0 0 0 Q =0 0 0 0
Quark confinement N =0 1 0 1
Lz = 0 0 0 Q =0 0 0 0
N =1 0 1 0
Lz = 1 0 1 Q =1 -1 1 -1
q--------------------q q--------------------q
Quark confinement N =0 1 0 1
Lz = 0 0 0 Q =0 0 0 0
N =1 0 1 0
Lz = 1 0 1 Q =1 -1 1 -1
N =1 1 0 0
Lz = 1 1 1 Q =1 0 0 -1
q--------------------q q--------------------q
q----------------------------------------------------------------------------q
Realization of plaquettes Method 1: (effective gauge invariance)
Unlike in 1+1 dimensional case, here one has to obtain plaquette terms using effective methods.
Imposing Gaussβs law as a constraint.
E. Zohar, BR, Phys. Rev. Lett. 107, 275301 (2011)
The abelian Kogut-Susskind model.
The βLoop Methodβ
1d elementary link interactions β already gauge invariant building blocks for effective plaquettes
Auxiliary fermions
E. Zohar, I. J. Cirac, BR, arXiv:1303.5040
Figure from ref [6]
The βLoop Methodβ
1d elementary link interactions β already gauge invariant building blocks for effective plaquettes
Auxiliary fermions
Virtual loops => plaquettes!
Figure from ref [6]
Confinement, flux breaking, glueballs
Flux loops deforming and breaking effects
Electric flux tubes
E. Zohar, BR, Phys. Rev. Lett. 107, 275301 (2011).
E. Zohar, J. I. Cirac, BR, Phys. Rev. Lett. 110, 055302 (2013)
Wilson loop measurments
Detecting Wilson Loopβs area law by interference of βMesonsβ.
E. Zohar, J. Ignacio Cirac, BR, PRL (2013).
E. Zohar , BR, New J. Phys. 15 (2013) 043041
III.) QCD: Non-abelian Yang-Mills gauge theory.
Non-abelian: Yang-Mills links
52 E. Zohar, I. J. Cirac, BR, arXiv:1303.5040
Each link has left and right degrees of freedom - SU(N) elements. The βrelative rotationβ corresponds to the nonabelian chagre on the link.
Non-Abelian: Yang Mills plaquettes
In this case, the plaquette terms take the
form
β1
2π2Trβ‘ πππ
β πβ + π». π.
Yang-Mills model.
Realizing Yang-Mills with atoms
β’ SU(2): 4 bosons on each link
54
E. Zohar, J. Ignacio Cirac, BR, Phys. Rev. Lett. 110, 125304 (2013) E. Zohar, I. J. Cirac, BR, arXiv:1303.5040
SU(2) in 1+1-d β’ Confinement of βcolorβ charges.
β’ Non-abelian Schwinger βQCD_2β
β’ Hamiltonian:
on each link β an SU(2) element ; conjugate to the groupβs left and right generators, and , satisfying (separately) SU(2) algebras
55
ZN Gauge theory
β’ Abelian discrete gauge theory: the gauge field degrees of freedom operate in a finite Hilbert space
β’ Three phases in 3+1 dimensions
Non-Conf.
Electric Conf. Magnetic Conf.
l
N
1
Nc
Adapted from Horn et. al., PRD 19, 3715, 1979
π βΌ πππΈ
π βΌ πππ΄
Simulating ZN Gauge theory
β’ Finite Hilbert spaces on links: one can realize unitary operators in the elementary link interactions, obtained using hybridized levels
β’ In a pure gauge theory, plaquettes are obtained similarly, using the βloop methodβ
E. Zohar, I. J. Cirac, BR, arXiv:1303.5040
Figure from ref [6]
Summary
I.) 1+1 D systems: Gauge invariant links
But given natural interactions
II.) Higher dimensions : We presented the loop method
For generating plaquette interactions
In 2+1 and higher D.
III.) Experiments: Probing Confinement,
pair Production (vacuum instability),
String breaking, string tension-
at a non perturbative regime! !
Gluons, Wilson loops, Phase transitionsβ¦.
FL FR
B1,B2
Figures from refs. [1,3,6]
Theory 1+1 Pure
1+1 with matter
d+1 Pure d+1 with matter
U(1) - cQED
Trivial K.S. and truncated
K.S. and truncated
K.S. and truncated
SU(2) βYang Mills
Trivial Full Simulation Strong limit Simulation
Strong limit Simulation
ZN Trivial
Summary and outlook
K.S. = Kogut Susskind Hamiltonian LGT
1. Confinement and Lattice Quantum-Electrodynamic Electric Flux Tubes Simulated with Ultracold Atoms Erez Zohar, BR Phys. Rev. Lett. 107, 275301 (2011) Preprint: arXiv: 1108.1562v2 [quant-ph]
2. Simulating Compact Quantum Electrodynamics with ultracold atoms: Probing confinement and nonperturbative effects Erez Zohar, J. Ignacio Cirac, BR Phys. Rev. Lett. 109, 125302 (2012) Preprint: arXiv: 1204.6574 [quant-ph]
3. Topological Wilson-loop area law manifested using a superposition of loops Erez Zohar, BR New J. Phys. 15 (2013) 043041 Preprint: arXiv:1208.1012 [quant-ph]
4. Simulating 2+1d Lattice QED with dynamical matter using ultracold atoms Erez Zohar, J. Ignacio Cirac, BR Phys. Rev. Lett. 110, 055302 (2013) Preprint: arXiv:1208.4299 [quant-ph]
5. Cold-atom quantum simulator for SU(2) Yang-Mills lattice gauge theory Erez Zohar, J. Ignacio Cirac, BR Phys. Rev. Lett. 110, 125304 (2013) Preprint: arXiv:1211.2241 [quant-ph]
6. Quantum simulations of gauge theories with ultracold atoms: local gauge invariance from angular momentum conservation Erez Zohar, J. Ignacio Cirac, BR Preprint: arXiv:1303.5040 [quant-ph]
Thank you for your attention !
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