QUANTUM SIMULATION OF LGT Robert Ott Institute for Theoretical - - PowerPoint PPT Presentation

QUANTUM SIMULATION OF LGT

Robert Ott

Cold Quantum Coffee 30.06.2020

Institute for Theoretical Physics Heidelberg

OUTLINE

ψn

ψn+ˆ

x

ψn+2ˆ

x

Un,ˆ

x

Un+ˆ

x,ˆ x

Quantum simulation of LGT

1 2

Lattice gauge theory & QS

QS of U(1) quantum link model

LGT & QS „And I'm not happy with all the analyses that go with just the classical theory, because nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.“

• Richard Feynman, `Simulating Physics with Computers’

E.g.: N Spins-1/2 on a chain

„Complexity“ N

. . .

~exp(N)

LGT & QS

„Complexity“ N ~exp(N)

Path integral: Again: Monte Carlo integration of:

Z Dφ e−S[φ]

Z Dφ eiS[φ]

Monte Carlo integration of:

Sign problem!

Z Dφ e−S[φ]

Z Dφ eiS[φ]

LGT & QS Real-time/out-of-equilibrium phenomena: Schwinger Pair Production Heavy Ion Collisions Thermalization Equilibrium phenomena: QCD phase diagram High Tc superconductivity . . . . . .

LGT & QS Quantum Simulation scheme: High energy theory, Quantum simulator Condensed matter, . . .

• 1. Initialization
• 2. Evolution
• 3. Read-out

Question Answer

Robert Ott, Institute for Theoretical Physics Heidelberg

Mil, Zache, Hedge, Xia, Bhatt, Oberthaler, Hauke, Berges and Jendrzejewski Science 367.6482 (2020): 1128-1130. Martinez, Muschik, Schindler, Nigg, Erhard, Heyl, Hauke, Dalmonte, Monz, Zoller and Blatt, Nature 534.7608 (2016): 516-519. Schweizer, Grusdt, Berngruber, Barbiero, Demler, Goldman, Bloch and Aidelsburger, Nature Physics 15.11 (2019): 1168-1173.

LGT & QS Notable experiments:

LGT & QS How to get QED into a quantum simulator? Identification of d.o.f. Lattice Gauge Theory Quantum Simulator

space, time

• ptical Lattice,

Hamiltonian time evolution gauge symmetry angular momentum conservation, mapping gauge fields bosonic atoms fermions fermionic atoms

Example for cold atoms: In general: Any kind of mapping, which (in a certain limit) reproduces the original theory.

effective theory

LGT & QS Identification Theorists Experimentalists (effective) action … path integral … QFT … Hilbert space … Hamiltonian … States …

?!?

LGT & QS Hamiltonian of QED in one (spatial) dimension? Start from lattice field theory:

ψn

ψn+ˆ

x

ψn+2ˆ

x

Un,ˆ

x

Un+ˆ

x,ˆ x

ψn → e−ieαnψn

Un,n+1 → eieαnUn,n+1eieαn+1

ψ†

n∂xψn

1 2aψ†

n (ψn+1 − ψn−1)

lattice Gauge transformations: i.e. breaks gauge symmetry!

1 2aψ†

n (ψn+1 − ψn−1)

Introduce gauge links: Discretize space: (for naive fermions here) as

1 2aψ†

n (Un,n+1ψn+1 − Un,n−1ψn−1)

[En,n+1, Um,m+1] = eδnmUm,m+1

electric field:

LGT & QS General gauge transformation:

V = ei P

n αnGn

ψn → e−ieαnψn

Un,n+1 → eieαnUn,n+1eieαn+1

O → V †OV

with

Gn = En,n+1 − En,n−1 − eaψ†

nψn

Gauss law: Gauge invariant Hamiltonian:

Gn|ψi = 0

Gn|ψi = qn|ψi

more generally: external charges Gauge invariant dynamics:

H → H

[H, Gn] = 0

i.e.

Gn|ψ(t)i = 0

H = 1 2 X

n

E2

n,n+1 + m

X

n

ψ†

nψn + 1

2a X

n

(ψ†

nUn,n+1ψn+1 + h.c.)

LGT & QS `Single link’ Hilbert space:

n

n + 1

n − 1

[E, U] = U

Electric field basis: E|Ei = E|Ei

U|Ei = |E + 1i

H = span{|Ei | 2 eZ}

Using : Hilbert space: |0i

|1i

|2i

|-2i

|-1i

E/e

Quantum link models (QLMs): |0i

|1i

|2i

|-2i

|-1i

E/e

new “class” of gauge theories

U → L+ p l(l + 1)

E → eLz

|0i

|1i

|-1i

E/e

|-li

|li

QED QLM LGT & QS

Chandrasekharan, S. & Wiese, U. J.

• Nucl. Phys. B 492, 455–471 (1997)

truncate

H = 1 2 X

n

(Lz

n,n+1)2 + m

X

n

ψ†

nψn +

1 2a p l(l + 1) X

n

(ψ†

nL+ n,n+1ψn+1 + h.c.)

OUTLINE

ψn

ψn+ˆ

x

ψn+2ˆ

x

Un,ˆ

x

Un+ˆ

x,ˆ x

Lattice gauge theory & QS

Quantum simulation of LGT

1 2

QS of U(1) quantum link model

(based on arXiv:2003.08945v1)

QS of U(1) QLM Spin-1/2 link: |1/2i

|-1/2i

Staggered fermions:

+

• |0i

|1i

|0i

|1i

even sites

• dd sites

+

Spin-1/2 QLM:

=

fermion no fermion

H = m X

n

ψ†

nψn + ˜

t X

n

(ψ†

nS+ n,n+1ψn+1 + h.c.)

+

• E.g.:

H = 1 2 X

n

(Lz

n,n+1)2 + m

X

n

ψ†

nψn +

1 2a p l(l + 1) X

n

(ψ†

nL+ n,n+1ψn+1 + h.c.)

QS of U(1) QLM Bose Hubbard model:

HBH = J X

i

(b†

ibi+1 + h.c.) +

X

i

✏ib†

ibi + U

2 X

i

b†

ib† ibibi

J

✏i

• ptical lattice

single bosonic atoms

QS of U(1) QLM Unit cell: Energy effective interaction integrate out effectively

Heff ∼ J2 U (b1b†

2b† 2b3 + h.c.) + m(b† 1b1 + b† 3b3)

+

• energy imbalance

1 3 1 3

δ − U 2

˜ t(ψ†

1S+ 2 ψ3 + h.c.)

Jordan Wigner transformation

QS of U(1) QLM Mapping to Bose-Hubbard model via Jordan Wigner transformation:

+

• e+ e- pair

even sites

• dd sites

vacuum vacuum

+

• |0i

|1i

even sites

• dd sites

|0i

|1i

boson no boson even sites

• dd sites

Gauss law

G = rE ρ = 0

QS of U(1) QLM Many-body system:

+ + +

• m → −∞

m → ∞

ground state at 2 degenerate ground states at dynamically tune mass parameter QCP

QS of U(1) QLM Experimental and numerical results:

Gi|ψi = 0

projects onto states with

(arXiv:2003.08945v1)

✏(t) = 1 1 N X

i

h (t)|Pi| (t)i

state population

Summary: Quantum simulation of LGT Non-equilibrium dynamics is hard — Quantum Simulation? We performed a quantum simulation of an extended U(1) gauge theory Extension to higher dimensions, more complicated symmetry groups, etc. Certification? How can we be sure the symmetry is there? What other platforms can be used?

THANK YOU FOR YOUR ATTENTION!

Torsten V. Zache (ITP) Jürgen Berges (ITP) Philipp Hauke (UniTN) Bing Yang (PI HD) Jad Halimeh (UniTN) Hui Sun (PI HD)