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Atomic Quantum Simulation of
(Non-)Abelian Lattice Gauge Theories:
Quantum Link Models
UNIVERSITY OF INNSBRUCK
Peter Zoller
Innsbruck
M. Dalmonte
E. Rico→ Ulm
M. Müller→ Madrid
D. Marcos
K. Stannigel
(quantum optics)
ITP Bern
D. Banerjee
M. Bögli
P. Stebler
U. J. Wiese
(high-energy physics)
IQOQI
AUSTRIAN ACADEMY OF SCIENCES
€U AQUTE
€U COHERENCE
UQUAM
AFOSR
Atomic Quantum Simulation of
(Non-)Abelian Lattice Gauge Theories:
Quantum Link Models
UNIVERSITY OF INNSBRUCK
Peter Zoller
… possible future developments at the interface between
AMO and cond mat & particle physics
•
•
•
Abelian U(1) Schwinger model, D. Banerjee et al., PRL 2012
IQOQI
AUSTRIAN ACADEMY OF SCIENCES
€U AQUTE
€U COHERENCE
U(N), SU(N) Non-Abelian QLM, D. Banerjee et al., PRL in print
UQUAM
Other
- dissipative techniques K. Stannigel et al.
- other implementations: superconducting qubits, D. Marcos et al.
AFOSR
Related work: MPQ-Tel Aviv, ICFO, Cornell, ...
… lattice gauge theories [in particle physics]
•
•
Gauge theories on a discrete lattice structure
K. Wilson, Phys. Rev. D (1974).
Fundamental gauge symmetries: standard model (every force has a
gauge boson)
non-perturbative approach to
fundamental theories of matter
(e.g. QCD)
→ classical statistical mechanics
a
quarks and gluons
lattice
Classical Monte Carlo simulations:
achievements
• first evidence of quark-gluon plasma
• ab initio estimate of proton mass
• entire hadronic spectrum
issues
Sign problem in its various flavors:
• finite density QCD (=fermions)
• real time evolution
Quantum simulation (with atoms)? … toy models & simple phenomena
3
Quantum Simulation of Lattice Gauge Theories with Atoms
Lattice Gauge Theories
Atomic Quantum Simulation
gauge
field
matter field
optical lattice
Hamiltonian formulation
Kogut & Susskind
(Non-)Abelian LGT:
✓QED U(1)
✓QCD SU(N), U(N)
gauge symmetry as
local symmetry
Hubbard models with bosonic and
fermionic atoms in optical lattices
?
“emergent lattice gauge theory”
Quantum Simulation of Lattice Gauge Theories with Atoms
Lattice Gauge Theories
gauge
field
matter field
(Non-)Abelian LGT:
✓QED U(1)
✓QCD SU(N), U(N)
Quantum Link Models (QLM)
Example:
U(1) / QED
spin S
fermionic atoms
as matter field as quantum link: “spin ice”
QLM
✓gauge fields in finite dim spaces
✓Abelian & Non-Abelian
particle physics: U.J. Wiese et al., Horn, Orland & Rohrlich, …
cond mat: … [spin ice, quantum spin liquids]
QLM have “natural” implementations
with cold atoms in optical lattices
5
•
Quantum Link Models (as Hubbard Models for Atoms)
U(1) Abelian LGT
D. Banerjee et al., PRL 2012
“spin ice” QED + matter
L R
spin S
as quantum link
fermionic atoms
as matter field
Schwinger boson
atomic boson-fermi mixtures in optical lattices
U(N),SU(N) Non-Abelian LGT
D. Banerjee et al., PRL in print
fermions
L R
i
x
(i = 1, 2, 3)
color index
fermionic rishons
multi-species fermi gases
6
H
gauge
invariant
subspace
⇢
“spin ice” QED + matter
r·E =0
Gauss law as a constraint
Hamiltonians, (local) gauge symmetries, Gauss law
etc.
an AMO perspective
7
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
c-number / static gauge fields
particles hopping around a plaquette acquire a finite phase
=
ˆ
flux
x
~ ⇥A
~
d2 f~ · r
H=
matter
field
t
y
† i'xy
xe
y
phase 'xy =
+ h.c.
ˆ
y
x
~
d~l · A
U(1) (abelian)
theory & AMO experiments
on “synthetic gauge fields”
- Hofstadter butterfly
- Fractional Quantum Hall,
Fractional Chern insulators
Review: J. Dalibard et al. Rev. Mod. Phys. (2011)
8
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
dynamical gauge fields
particles hopping around a plaquette assisted by link degrees of freedom
gauge
field
x
H=
t
y
†
x Uxy
y
+ h.c. + . . .
link operator
matter field
9
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
dynamical gauge fields
particles hopping around a plaquette assisted by link degrees of freedom
Gx
x
H=
gauge bosons
fermions
unitary trafo:
V
! ei↵x Uxy e
i↵y
V
x
! ei↵x
V =
†
x Uxy
y
Gy
+ h.c. + . . .
local conserved quantity
gauge (local) symmetry U(1)
Uxy
t
y
Y
x
x
ei↵x Gx
generator
,
[H, Gx ] = 0 8x
generator of gauge transformation
Static vs. Dynamical Gauge Fields on Lattices: U(1)
•
dynamical gauge fields
particles hopping around a plaquette assisted by link degrees of freedom
x
H=
Gauss Law
Gx =
†
x
x
matter
⇢
X⇣
r·E =0
i
Ex,x+î
Ex
î,x
⌘
electric field operator
t
y
†
x Uxy
y
+ h.c. + . . .
local conserved quantity
[H, Gx ] = 0 8x
generator of gauge transformation
QED as “Quantum Spin Ice” [Quantum Link Model]
+
Ux,x+1 ! Sx,x+1
z
Ex,x+1 ! Sx,x+1
electric flux
Spin S=½,1,...
spin S
as quantum link
quantum link carrying an electric flux
spin-1/2
E = +1/2
E=
1/2
spin-1
E = +1
E=0
E=
1
12
QED as “Quantum Spin Ice” [Quantum Link Model]
+
Ux,x+1 ! Sx,x+1
z
Ex,x+1 ! Sx,x+1
electric flux
Spin S=½,1,...
spin S
as quantum link
configurations: spin-1/2
Gauss Law
charge = +1
+
⇢
...
charge = 0
“ice rule”
r·E =0
13
QED as “Quantum Spin Ice” [Quantum Link Model]
+
Ux,x+1 ! Sx,x+1
z
Ex,x+1 ! Sx,x+1
electric flux
Spin S=½,1,...
spin S
as quantum link
configurations: spin-1/2
H
+
physical
subspace
G x |√i = 0
8x
...
≥
¥
3
3
“ice rule” G x = ßµ S x°µ̂,µ ° S x,µ $ r · E = 0
14
A first remark on implementation: enforcing “Gauss law”
•
•
Lattice Gauge Theory: gauge symmetry fundamental
Implementation: gauge symmetry approximate → protect against errors
Strategies for Microscopic Implementation
H
☹
☺
G x |√i = 0
Hamiltonian
+ constraints G x |√i = 0
8x
8x
physical gauge
invariant subspace
15
A first remark on implementation: enforcing “Gauss law”
•
•
Lattice Gauge Theory: gauge symmetry fundamental
Implementation: gauge symmetry approximate → protect against errors
Strategies for Microscopic Implementation
H
1. Energy Constraints (as in cond mat)
energy
Hmicro = U
X
x
G x2 + . . .
U (large)
G x |√i = 0
✓interaction
✓emergent lattice gauge theory
see example below
...
...
16
A first remark on implementation: enforcing “Gauss law”
•
•
Lattice Gauge Theory: gauge symmetry fundamental
Implementation: gauge symmetry approximate → protect against errors
Strategies for Microscopic Implementation
H
2. “Classical Zeno effect”
K. Stannigel
17
A toy model: “motional narrowing”
classical noise
Two-level atom +
dephasing
|ei
Rabi ⌦
frequency
Fermi’s Golden Rule
classical
wiggling
laser
|gi
Compare “Zeno effect” by losses
in optical lattices
Pg (t) = exp
✓
! 1 for
⌦
2
t
◆
!1
population frozen in ground state
Rempe, Cirac et al.
Daley et al.
18
A first remark on implementation: enforcing “Gauss law”
•
•
Lattice Gauge Theory: gauge symmetry fundamental
Implementation: gauge symmetry approximate → protect against errors
Strategies for Microscopic Implementation
H
2. “Classical Zeno effect” K. Stannigel
Hmicro =
X
x
ªx (t )G x + . . .
see example below
✓white noise
✓linear in G ~ single particle terms ☺
Rem.: decoherence free subspace, bang-bang
19
Example 1:
The simplest (meaningful) quantum link model:
1D Schwinger model
AMO Implementation:
Bose-Fermi Mixtures in Optical Lattices
string breaking
Boson (gauge field)
...
...
Fermion (quark)
“quantum spin ice coupled to matter”
D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, U. J. Wiese, and P.Z., PRL 2012
20
Schwinger Model: U(1) fermions + gauge bosons in 1D
•
Hamiltonian: staggered fermions in 1D coupled to quantum link spin S
H
=
i
Xh †
X
g2 X 2
E x,x+1 ° t
√x U x,x+1 √x+1 + h.c. + m (°1)x √†x √x
2 x
x
x
electric flux
hopping
staggered fermions
Boson (gauge field)
...
...
Fermion (quark)
21
Schwinger Model: U(1) fermions + gauge bosons in 1D
•
Hamiltonian: staggered fermions in 1D coupled to quantum link spin S
H
=
g2 X z
(S x,x+1 )2
2 x
electric flux
t
X⇥
† +
x Sx,x+1
x
hopping
x+1
⇤
+ h.c. + m
X
( 1)x
†
x
x
x
staggered fermions
Boson (gauge field)
...
...
Fermion (quark)
22
Schwinger Model: U(1) fermions + gauge bosons in 1D
•
Hamiltonian: staggered fermions in 1D coupled to quantum link spin S
H
=
g2 X z
(S x,x+1 )2
2 x
electric flux
t
X⇥
† +
x Sx,x+1
x+1
⇤
+ h.c. + m
x
X
( 1)x
†
x
x
x
staggered fermions
hopping
...
...
Staggered fermions:
energy
empty
mass gap
mass gap
0
1
2
filling = ½
...
filled
fermi sea
23
Schwinger Model: U(1) fermions + gauge bosons in 1D
•
Hamiltonian: staggered fermions in 1D coupled to quantum link spin S
H
=
g2 X z
(S x,x+1 )2
2 x
electric flux
t
X⇥
† +
x Sx,x+1
x+1
⇤
+ h.c. + m
x
X
( 1)x
†
x
x
x
staggered fermions
hopping
...
...
Staggered fermions:
energy
q
mass gap
mass gap
0
1
2
filling = ½
q̄
...
24
Implementation with Atoms: Hamiltonian
•
building blocks
x
y
Spin S=½,1,...
25
Implementation with Atoms: Hamiltonian
x
y
Spin S=½,1,...
•
Spin as Schwinger bosons
L
R
L
R
bosons
N bosons in double well (S=N/2)
+
Sx,y
= b†L bR
⇣
1
z
Sx,y
=
b†R bR
2
b†L bL
⌘
Hamiltonian
hB =
+
z
tB (Sx,y
+ h.c.) + UB (Sx,y
)2
hopping
electric energy
26
Implementation with Atoms: Hamiltonian
x
H=
•
y
t
† +
x Sxy
y
+ h.c.
correlated hopping
L
R
L
R
bosons
interaction
fermions
H=
tB tF
U
† †
x bR bL
L
R
fermions as matter fields
y
+ h.c.
bosons as link variable
27
Gauss Law (as a constraint)
•
Gauss’ law
G x = √†x √x
+
(°1)x °1
2
° (E x,x+1 ° E x°1,x )
G x | physical statesi = 0 √! Ω ° r · E = 0
•
Example: Spin 1 representation
|0i
|1i
Even sites
|
1i
|0i
| + 1i
Odd sites
28
Implementation with Atoms: Gauss Constraints
•
Bose-Fermi mixtures in superlattices
tB
U↵
bosons
fermions
bosons
•
tF
Gauss constraint
1
2
G̃x = nF
+
n
+
n
x
x
x
2S +
1
[( 1)x
2
1]
~ total number of atoms on site x fixed: “super-Mott insulator”
29
Implementation with Atoms: Gauss Constraints
•
Bose-Fermi mixtures in superlattices
tB
U↵
bosons
fermions
bosons
•
tF
enforcing the Gauss Law as an energy constraint
energy
Hmicroscopic = U
X
G̃2x + . . .
x
Bose + Fermi Hubbard model
•
U (large)
G̃x |physical statesi = 0
emergent lattice gauge theory
- dynamics in physical subspace: analogous to t-J model
- we have verified the reduction: microscopic to the
quantum link model at the few- and many-body level
30
43
String breaking and confinement
string breaking
Q̄
Q
Unbroken
String
E string ° E 0 =
Mesons +
vacuum
g2
2 (L ° 1)
E mesons ° E 0 = 2
Critical string length
≥
g2
2
+m
¥
L(c) = 2 + 2m/g 2
We have verified this dynamics in the microscopic model
Rem.: string breaking 1D vs. 3D / non-Abelian
31
Example 2:
H
classical
noise
Enforcing Gauss’ Law by Classical Noise
K. Stannigel
32
K. Stannigel
1D Schwinger Model: Fermions → Spins
•
Hamiltonian
Sx,x+1
...
...
x+1
x
H0 =
J
3
X
x Sx,x+1
+
x+1
+ h.c. + m(t)
x=1
•
4
X
( 1)x
z
x
x=1
Gauss Law
Gx =
Sxz 1,x
+
z
x
z
Sx,x+1
1
+ ( 1)x
2
33
H
• We add a gauge-variant term to the
Hamiltonian ...
H1 =
3
X
x
+
x+1
+ h.c.
x=1
imperfection
system leaves
Gauge invariant
subspace
1
0.8
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
50
100
150
200 250
Jt
300
350
imperfection
noise
400
0
0
50
100
150
200 250
Jt
300
350
400
• … and suppress it by noise
K. Stannigel
34
y
x
cix cj†
y
Non-Abelian U(N) and SU(N) Quantum Link
Models with Fermionic Atoms
… a few basic aspects
D. Banerjee, M. Bögli, M. Dalmonte, E. Rico, P. Stebler, U. J. Wiese, & PZ,
PRL in print.
[Innsbruck - Bern]
35
Bern
Innsbruck
Non-Abelian Lattice Gauge Theory
•
Example: U(N), SU(N), …
fermions as matter fields
i
x
non-Abelian gauge fields
ij
Ux,y
(i = 1, . . . , N )
• NxN unitary matrices
• generators
(color)
[
H=
t
N
X
i† ij
x Uxy
j
y
a
ij
a
,
(a = 1, . . . N 2
b
] = i2fabc
1)
c
+ h.c. + . . .
i,j=1
36
Bern
Innsbruck
Non-Abelian Lattice Gauge Theory
•
Quantum Link Models
L
R
U ij = ciR cj†
L
(i = 1, . . . , N )
representation as fermionic rishons
•
Abelian U(1)
L
fermions as matter fields
R
bosons as link variable
37
Bern
Innsbruck
Non-Abelian Lattice Gauge Theory
•
Quantum Link Models
L R
quark
H=
rishon
0
!
N
N
X
X
j†
i† i
@
c
t
c
x R
L
i=1
j=1
1
jA
y
+ h.c.
(separable interaction ☺)
Implementation
✓ fermionic atoms with N internal states (color)
38
Bern
Innsbruck
Non-Abelian Lattice Gauge Theory
•
Quantum Link Models
L R
H=
0
!
N
N
X
X
j†
i† i
@
c
t
c
x R
L
i=1
j=1
1
jA
y
+ h.c.
(separable interaction ☺)
Dynamics
✓ fermionic atoms converts itself from quark to rishon & vice versa
✓correlated hop
39
Bern
Innsbruck
Non-Abelian Lattice Gauge Theory
•
Quantum Link Models
L R
quark
H=
rishon
0
!
N
N
X
X
j†
i† i
@
c
t
c
x R
L
j=1
i=1
1
jA
y
+ h.c.
(separable interaction ☺)
“nuclear and particle physics”
a a
a†
g
nucleon
a a
g g
glueball
a
meson
triton
g
We know how to
implement gauge groups
SU(2), SU(3), SO(3), …
[at the moment in 1D]
PRL in print
deuteron
40
Implementation: enforcing Gauss Law / local symmetry
•
strategy 3:
U(N),SU(N) Non-Abelian LGT
implementation
fundamental local symmetry /
conservation law
M. Dalmonte
elementary
building block
local particle # conservation
(implemented with high precision)
fermions
41
Summary
U(1) Abelian LGT
U(N),SU(N) Non-Abelian LGT
L R
L R
atomic boson-fermi mixtures
D. Banerjee et al., PRL in print
H
multi-species fermi gases
D. Banerjee et al., PRL 2012
Hamiltonian + Gauss constraint:
1.Energy constraints
2.Classical Noise
physical gauge
invariant subspace
3.Microscopic Hamiltonian is gauge invariant
We are on our way to find simpler, and thus more realistic implementations
42
The Collaboration
•
IQOQI - Innsbruck University
M. Dalmonte
•
E. Rico
D. Marcos
•
K. Stannigel
Ibk→ Madrid
M. Müller
Albert Einstein Center - Bern University
D. Banerjee
M. Bögli
P. Stebler
P. Widmer
U.-J. Wiese
43
Outlook
•
gauge fields in 2D, 3D
y
x
•
cix cj†
y
ij
Ux,y
superconducting qubits
cavity a
y
x
cavity b
JJ / SQUID
1
qubit 1
qubit 1
cavity b
cavity a
qubit 2
qubit 2
D. Marcos et al., unpublished
•
2
21
Is there life after optical lattices ...
44
Nano-Scale Trap Arrays from Superconducting
Vortices
O. Romero-Isart, C. Navau, A. Sanchez, P. Zoller, and J. I. Cirac, arXiv 2013
Oriol Romero-Isart
•
a
Pinned vortices in type-II superconductors as magnetic trap arrays
- lattice spacings down to 50 nm
- nano-structuring of surfaces & holes
b
B1
z
BM(t)
0
z0
dm
a
Superconducting Vortex Lattice for Atoms
Mesocopic size magnetic / superconducting chip traps (Zimmermann, Dumke,
Haroche, …)
45