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

• Condensed Matter Models ( high TC?) Artificial potentials (Topological insulators, Hall effect) • Gravity: Black hole Hawking radiation, quantum effects of cosmological expansion. • QFT: Scalar and fermion relativistic fields,

• Condensed Matter Models ( high TC?) Artificial potentials (Topological insulators, Hall effect) • Gravity: Black hole Hawking radiation, quantum effects of cosmological expansion. • QFT: Scalar and fermion relativistic fields,

• .

• High Energy physics (HEP) • Quantum Chromodynamics (QCD) effects? • Confinement of Quarks?

Quantum simulations

Simulating systems

Using one controllable quantum system in order to simulate another system,

for example by:

• BECs • Atoms in optical lattices • Rydberg Atoms • Trapped Ions • Superconducting devices • …

4

Simulating HEP Physics

• Experimentally more challenging compared with cond. mat. simulations.

• Could be used to explore otherwise experimentally and computationally inaccessible effects, and directly unobservable phenomena at short scales.

• 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 – 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.

• Outlook and Summary

Requirements: HEP models

I) fields:

Fermion fields := Matter

Bosonic gauge fields := Interaction mediators

II) local gauge invariance => “charge” conservation (exact, or low energy, effective )

III) Relativistic invariance => causal structure ( In the continuum limit.)

=> dynamics and interactions

Structure of HEP models

• Matter particles:= fermions (+Higgs) (Charges: electric, color , flavor. Spin. Mass)

• Gauge fields:= mediate the Interactions

Electromagnetic: massless photon, (1), U(1)

Weak interaction : massive Z, W’s , (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

𝛼𝑄𝐸𝐷 ≪ 1, 𝑉𝑄𝐸𝐷 𝑟 ∝1

𝑟

We (ordinarily) don’t need second quantization to understand

the structure of atoms: 𝑚𝑒𝑐2 ≫ 𝐸𝑅𝑦𝑑𝑏𝑒𝑟𝑔 ≃ 𝛼𝑄𝐸𝐷

2 𝑚𝑒𝑐2

Abelian: QED

𝛼𝑄𝐸𝐷 ≪ 1, 𝑉𝑄𝐸𝐷 𝑟 ∝1

𝑟

We (ordinarily) don’t need second quantization to understand

the structure of atoms. 𝑚𝑒𝑐2 ≫ 𝐸𝑅𝑦𝑑𝑏𝑒𝑟𝑔 ≃ 𝛼𝑄𝐸𝐷

2 𝑚𝑒𝑐2

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:

…+

+ +…

( 891) Feynman vertex diagrams)

(g-2)/2= ….

…+

12672 self energy diagrams

(g-2)/2= 1 159 652 180.73 (0.28) × 10−12 g − 2 measurement by the Harvard Group using a Penning trap

T. Aoyama et. al. Prog. Theor. Exp. Phys. 2012, 01A107

Non-abelian: QCD

𝛼𝑄𝐶𝐷 > 1 , 𝑉𝑄𝐶𝐷 𝑟 ∝ 𝑟

non-perturbative confinement effect!

=> structure of Hadrons: quark pairs form Mesons , quark triplets form Baryons. Color Electric flux-tubes: “a non-abelian Meissner effect”.

Non-abelian: QCD

𝛼𝑄𝐶𝐷 > 1 , 𝑉𝑄𝐶𝐷 𝑟 ∝ 𝑟

non-perturbative confinement effect!

=> structure of Hadrons: quark pairs form Mesons , quark triplets form Baryons. Color Electric flux-tubes: “a non-abelian Meissner effect”.

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 • The standard avenue: 𝑍𝑙𝑎𝑡𝑡𝑖𝑐𝑒 − computed using

Monte Carlo “sampling” (Other methods exist: non-abelian bosonization (1+1); instanton semi-classical summation, large N perturbation, variational methods… )

------------------

… but:

• Correlations but not time dependence

• Limited applicability with too many quarks .

Computationally hard “sign problem”.

(e.g. color superconductivity, Quark-Gluon plasma. )

• Quark confinement with dynamic matter never proven.

First step:

• Confinement in Abelian lattice models!

• Toy models with “QCD-like” properties that capture the essential physics mechanism

of confinement.

Confinement in abelian compact QED lattice models

• 1+1D: “Schwinger’s model” manifests confinement. (analytic and lattice results available).

• 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.)

• 3+1D: phase transition between strong coupling

confinement phase, and weak coupling coulomb phase.

• Z_N discrete gauge symmetry: in d+1, d>1. Electric and magnetic confinement & deconfinement

QED in 1+1 : Schwinger’s model

• No magnetic fields: EM has no dynamics of its own. Non trivial dynamics obtained by coupling to dynamical charge sources.

• Schwinger: 𝑒+𝑒− form bound states. (analytic and lattice results available.)

• Non-abelian extension: in 1+1: Q𝐶𝐷2 version, not completely solved. Only in the large-N limit.

1+1, U(1) gauge theory

𝐻 = 𝑀𝑛𝜓𝑛†𝜓𝑛

𝑛

+ 𝛼𝑛 𝜓𝑛†𝜓𝑛+1 +𝐻. 𝑐.

Start with a hopping fermionic Hamiltonian, in 1 spatial direction

This Hamiltonian is invariant to global gauge transformations,

𝜓𝑛⟶ 𝑒−𝑖Λ𝜓𝑛 ; 𝜓𝑛

†⟶ 𝑒𝑖Λ𝜓𝑛†

1+1, U(1) gauge theory

𝐻 = 𝑀𝑛𝜓𝑛†𝜓𝑛

𝑛

+ 𝛼𝑛 𝜓𝑛†𝑈𝑛𝜓𝑛+1 +𝐻. 𝑐.

Promote the gauge transformation to be local:

Then, in order to make the Hamiltonian gauge invariant, add unitary operators, 𝑈𝑛,

𝜓𝑛⟶ 𝑒−𝑖Λ𝑛𝜓𝑛 ; 𝜓𝑛

†⟶ 𝑒𝑖Λ𝑛𝜓𝑛†

𝑈𝑛 = 𝑒𝑖𝜃𝑛 ; 𝜃𝑛⟶ 𝜃𝑛+ Λ𝑛+1 - Λ𝑛

(analogous to 𝐴 → 𝐴 + ∇ Λ )

Dynamic 1+1 U(1) gauge theory

Add dynamics to the gauge field:

Where 𝐿𝑛 is the angular momentum operator

conjugate to 𝜃𝑛 , representing the (integer) electric field.

𝐻𝐸 =𝑔2

2 𝐿𝑛,𝑧

2

𝑛

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 𝛻 × 𝑨 2 - magnetic energy term.

− 1𝑔2 cos 𝜃𝑚,𝑛

1 + 𝜃𝑚+1,𝑛2 − 𝜃𝑚,𝑛+1

1 − 𝜃𝑚,𝑛2

𝑚,𝑛

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: 𝐴 → 𝜃 , 𝐸 → 𝐿

Matter fields: 𝜓, 𝜓†

Atomic analog simulator

Figure from ref [6]

General idea of our work

• Links: realized by atomic scattering : gauge invariance is

fundamental

• 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

𝐿 → 𝐿 − 1

F

Fermion

FL FR

B1,B2

F

Fermion

FL FR

B1,B2

𝐿 → 𝐿 + 1

F

Fermion

Figure from ref [6]

Fermionic atom Bosonic atoms

Angular Momentum conservation Local gauge invariance

FL FR

B1,B2

𝜓𝐿†𝑏1†𝑏2𝜓𝑅 + 𝜓𝑅

†𝑏2†𝑏1𝜓𝐿

mF (FL)

mF (FR)

mF (B1)

mF (B1)

Angular Momentum conservation Local gauge invariance

FL FR

B1,B2

𝜓𝐿†𝑏1†𝑏2𝜓𝑅 + 𝜓𝑅

†𝑏2†𝑏1𝜓𝐿

mF (FL)

mF (FR)

mF (B1)

mF (B1)

Angular Momentum conservation Local gauge invariance

FL FR

B1,B2

𝜓𝐿†𝑏1†𝑏2𝜓𝑅 + 𝜓𝑅

†𝑏2†𝑏1𝜓𝐿

mF (FL)

mF (FR)

mF (B1)

mF (B1)

Gauge U(1) bosons: Schwinger algebra

𝐿+ = 𝑏1†𝑏2 ; 𝐿− = 𝑏2

†𝑏1

𝐿𝑧 =1

2𝑁1 − 𝑁2 ;𝑙 =

1

2𝑁1 +𝑁2

Gauge bosons: Schwinger algebra

𝐿+ = 𝑏1†𝑏2 ; 𝐿− = 𝑏2

†𝑏1

𝐿𝑧 =1

2𝑁1 − 𝑁2 ; 𝑙 =

1

2𝑁1 +𝑁2

and thus what we have is

𝜓𝐿†𝐿+𝜓𝑅 + 𝜓𝑅

†𝐿−𝜓𝐿

𝜓𝐿†𝑏1†𝑏2𝜓𝑅 + 𝜓𝑅

†𝑏2†𝑏1𝜓𝐿

Gauge bosons: Schwinger algebra

𝜓𝐿†𝐿+𝜓𝑅 + 𝜓𝑅

†𝐿−𝜓𝐿

For large 𝑙 , 𝑚 ≪ 𝑙

𝐿+ = 𝑏1†𝑏2~𝑒

𝑖 𝜙1−𝜙2 ≡ 𝑒𝑖𝜃 = 𝑈

𝜓𝐿†𝑈𝜓𝑅 + 𝜓𝑅

†𝑈†𝜓𝐿

Qualitatively similar results can be obtained with just two bosons on the link.

Electric dynamic term

𝐸2 = 𝐿𝑧2 =1

4𝑁1 −𝑁2

2

=1

4𝑏1†𝑏12+1

4𝑏2†𝑏22−1

2𝑏1†𝑏1𝑏2

†𝑏2

𝐻𝐸 =𝑔2

2 𝐿𝑛𝑧

2

𝑛

𝐸 = ( 𝑁1 − 𝑁2)/2 𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒 𝑡𝑜 𝜃 = 𝜙1 − 𝜙2

Finally: cQED; U(1) in 1+1

𝐻 = 𝑀 −1 𝑛𝜓𝑛†𝜓𝑛

𝑛

+ 𝛼 𝜓𝑛†𝑈𝑛𝜓𝑛+1 +𝐻. 𝑐.

+𝑔2

2 𝐿𝑛𝑧

2

𝑛

B-B Scattering: EM kinetic energy

F-B scattering: link interaction

The Schwinger model:

(“c”=“comact”)

Mass and Charge

• Even n (particles):

𝑄 = 𝑁 = 𝜓†𝜓 0 atoms: zero mass, zero charge

1 atom: M, Q=1

• Odd n (anti-particles):

𝑄 = 𝑁 − 1 1 atom : zero mass, zero charge (“Dirac sea”)

0 atoms: mass M (relative to –M), charge Q=-1

note: mass is measured relative to –M

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 Gauss’s law as a constraint.

E. Zohar, BR, Phys. Rev. Lett. 107, 275301 (2011)

The abelian Kogut-Susskind model.

The “Loop Method”

1d elementary link interactions – already gauge invariant building blocks for effective plaquettes

Auxiliary fermions

E. Zohar, I. J. Cirac, BR, arXiv:1303.5040

Figure from ref [6]

The “Loop Method”

1d elementary link interactions – 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 Loop’s area law by interference of “Mesons”.

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 “relative rotation” corresponds to the nonabelian chagre on the link.

Non-Abelian: Yang Mills plaquettes

In this case, the plaquette terms take the

form

−1

2𝑔2Tr□ 𝑈𝑈𝑈

†𝑈† + 𝐻. 𝑐.

Yang-Mills model.

Realizing Yang-Mills with atoms

• 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 • Confinement of “color” charges.

• Non-abelian Schwinger “QCD_2”

• Hamiltonian:

on each link – an SU(2) element ; conjugate to the group’s left and right generators, and , satisfying (separately) SU(2) algebras

55

ZN Gauge theory

• Abelian discrete gauge theory: the gauge field degrees of freedom operate in a finite Hilbert space

• 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

𝑃 ∼ 𝑒𝑖𝐸

𝑄 ∼ 𝑒𝑖𝐴

Simulating ZN Gauge theory

• Finite Hilbert spaces on links: one can realize unitary operators in the elementary link interactions, obtained using hybridized levels

• In a pure gauge theory, plaquettes are obtained similarly, using the “loop method”

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 transitions….

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) –Yang 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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