of a 1D lattice gauge theory
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Transcript of of a 1D lattice gauge theory
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Philipp Hauke, David Marcos, Marcello Dalmonte, Peter Zoller (IQOQI, Innsbruck)
Brighton, 18.12.2013
Phys. Rev. X 3, 041018 (2013)
Experimental input:Christian Roos, Ben Lanyon, Christian Hempel, René Gerritsma, Rainer Blatt
with trapped ions
Quantum simulation
of a 1D lattice gauge theory
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Gauge theories describe fundamental aspects of Nature
QCD
Spin liquids
Kitaev’s toric code is a gauge theory
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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Gauge theory
Physical states obey a local symmetry.
E.g.: Gauss’ law
In quantum mechanics, the gauge field acquires its own dynamics.
This symmetry couples kinetic terms to field
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To make amenable to computation gauge theory lattice gauge theory
Gauss’ law
K. Wilson, Phys. Rev. D 1974
Bermudez, Schaetz, Porras, 2011,2012Shi, Cirac 2012static gauge field
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To make it simpler, discretize also gauge field (quantum link model). Kogut 1979,Horn 1981, Orland, Rohrlich 1990, Chandrasekharand, Wiese 1997, Recent Review: U.-J. Wiese 2013
42S1/2
32D5/2| >| >
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For trapped-ion implementation:transform to spins (Jordan-Wigner)
Dynamics
Gauss’ law
Spins can be represented by internal states.
42S1/2
32D5/2| >| >
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Want to implement
Dynamics
Conservation law (Gauss’ law)
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Interesting phenomena in 1D QED
Hebenstreit et al., PRL 111, 201601 (2013)
time
dist
ance
string breaking
Charge density
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q qq–q–m/J→–∞ m/J→+∞
False-vacuum decayquark picture
spontaneously breaks charge and parity symmetry
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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Want to implement
Dynamics
Conservation law (Gauss’ law)
Rotate coordinate system
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gauge violating
Energy penalty protects Gauss’ law
total Hilbert space gauge
invariant
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Energy penalty protects Gauss’ law
spin-spin interactions
longitudinal field
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Need spin-spin interactions with equal strength between nearest- and next-nearest neighbors
Want Know how to do
Various experimentsSchaetz, Monroe, Bollinger, Blatt, Schmidt-Kaler, Wunderlich
TheoryPorras and Cirac, 2004Sørensen and Mølmer, 1999
See also Hayes et al., 2013Korenblit et al., 2012
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A closer look at the internal level structure
ΩσΩS
ΔEZee,D
ΔEZee,S42S1/2
32D5/2| >σ
| >σ
| >S
| >S
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Need spin-spin interactions with equal strength between nearest- and next-nearest neighbors
Want Know how to do
Solution:Use two different qubits to reinforce NNN interactions
+ dipolar tails
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Interactions protect gauge invariance.And allow to generate the dynamics!
2nd order perturbation theory
gauge violating
gauge invariant
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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q qq–q–
m/J→–∞ m/J→+∞False vacuum decay
quark picture
spin picture
breaks charge and parity symmetry
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A numerical test validates the microscopic equations
Perturbation theory valid
Dipolar tails negligible
P. Hauke, D. Marcos, M. Dalmonte, P. Zoller PRX (2013)
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Sweeps in O(1ms) reproduce the dynamics of the LGT
fidelity after quench
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S12σ1 σ2– + ––2+S21
A simpler proof-of-principle experiment with four ionsAvoids the need for fast-decaying interactions
Enforcing of Gauss law
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S12σ1 σ2+ –2+
–1/2S21
A simpler proof-of-principle experiment with four ionsAvoids the need for fast-decaying interactions
Remember interactions – –Use mode with amplitudes
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A simpler proof-of-principle experiment with four ionsAvoids the need for fast-decaying interactions
And does not suffer from dipolar errors S12σ1
σ2+ –2+
–1/2S21– –
–4 –2 0 2 4m/J –4 –2 0 2 4m/J
Compare scalable setup
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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Outline
One dimensional quantum electrodynamics
Trapped-ion implementationProposed schemeNumerical results
Protection of quantum gauge theory by classical noise
Conclusions
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gauge violating
Until now:Energetic protection.
total Hilbert space gauge
invariant
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Until now:Energetic protection.
For more complicated models, may require complicated and fine-tuned interactions
If we could do this with single-particle terms,
that would be much easier!
gauge # theory generatorsU(1) 1U(2) 4…
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Dissipative protection
white noise
→ Master equation
before
Stannigel et al., arXiv:1308.0528 (2013)
single-particle terms !
Gauge-invariant states are not disturbed
U(1) :
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Analogy:
driven two-level system + dephasing noise remains in ground state forever.
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gauge violating
gauge invariant
Problem: Cannot obtain dynamics as second-order perturbation
In neutral atoms, we found a way using intrinsic collisions.Stannigel et al., arXiv:1308.0528 (2013)
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ConclusionsProposal for a simple lattice gauge theory.
Ingredients:– Two different qubits (matter and gauge fields)– Two perpendicular interactions (one stronger than the other and fast decaying with distance)
– Single-particle terms
Numerics validate the microscopic Hamiltonian.– Statics– Dynamics (adiabatic sweep requires reasonable times)
A simpler proof-of-principle is possible with four ions.
| >| >| >
| >
S21
Phys. Rev. X 3, 041018 (2013)arXiv:1308.0528 (2013)
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Outlook
Implementations with higher spins or several “flavors.”
“Pure gauge” models in 2D.
Gauge invariance protected by the classical Zeno effect?arXiv:1308.0528
Optical latticesBanerjee et al., 2012, 2013 Tagliacozzo et al., 2012, 2013Zohar, Cirac, Reznik, 2012, 2013Kasamatsu et al., 2013
Superconducting qubitsMarcos et al., 2013
Static gauge fieldsBermudez, Schaetz, Porras, 2011, 2012Shi, Cirac, 2012
High-energy physics in ionsGerritsma et al, 2010 (Dirac equation)Casanova et al., 2011 (coupled quantum fields) Casanova et al., 2012 (Majorana equation)
Thank you !