Atomic Quantum Simulation of Abelian and non-abelian Gauge Theories

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1 Atomic Quantum Simulation of Abelian and non-abelian Gauge Theories Uwe-Jens Wiese Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Bern University Cold Atoms meet Quantum Field Theory 595. WE-Heraeus-Seminar Bad Honnef, July 8, 2015

2 Bern-Innsbruck Particle Physics AMO Collaboration Innsbruck: Marcello Dalmonte, Catherine Laflamme, Peter Zoller Bilbao: Enrique Rico Ortega Bern: Michael Bögli, Pascal Stebler, Philippe Widmer Berlin: Debasish Banerjee

3 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

4 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

5 The Standard Model a relativistic quantum field theory

6 The Standard Model a relativistic quantum field theory SU(3) Quantum Chromodynamics (QCD) Quarks Gluon

7 The Standard Model a relativistic quantum field theory SU(3) Quantum Chromodynamics (QCD) Quarks Baryon Gluon

8 The Standard Model a relativistic quantum field theory SU(3) Quantum Chromodynamics (QCD) Quarks Baryon Nucleus Gluon

9 Wilson s lattice Quantum Chromodynamics (QCD) verifies confinement of quarks and gluons inside protons and neutrons and confirms the experimentally observed hadron spectrum

10 Can heavy-ion collision physics or nuclear astrophysics benefit from quantum simulations in the long run?

11 Can heavy-ion collision physics or nuclear astrophysics benefit from quantum simulations in the long run?

12 Can heavy-ion collision physics or nuclear astrophysics benefit from quantum simulations in the long run?

13 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

14 The spin 1 quantum Heisenberg model 2 Quantum spins [S a x, S b y ] = iδ xy ε abc S c x and their Hamiltonian H = J xy Sx S y Partition function at inverse temperature β = 1/T Z = Tr exp( βh)

15 Low-energy effective action for antiferromagnetic magnons β S[ e] = dt d 2 x ρ ( s i e i e + 1 ) 2 c 2 t e t e 0 Fit to analytic predictions of effective theory 12 χ s ρ s = (11)J c = (3)Ja M s = (1)/a 2 χ 2 /d.o.f. = 1.15 L = 48a L = 56a L = 64a L = 72a L = 80a L = 88a L = 96a < W t 2 > ρ s = (11)J c = (3)Ja χ 2 /d.o.f. = 1.15 L = 48a L = 56a L = 64a L = 72a L = 80a L = 88a L = 96a M s = (1)/a βj βj M s = (1), ρ s = (11)J, c = (3)Ja UJW, H.-P. Ying (1994); F.-J. Jiang, UJW (2010)

16 Optical lattice quantum simulation of quantum spin systems J. Simon, W. S. Bakir, R. Ma, M. E. Tal, P. M. Preis, M. Greiner, Nature 472 (2011) 307.

17 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

18 Different descriptions of dynamical Abelian gauge fields: Maxwell s classical electromagnetic gauge fields E( x, t) = ρ( x, t), B( x, t) = 0, B( x, t) = A( x, t)

19 Different descriptions of dynamical Abelian gauge fields: Maxwell s classical electromagnetic gauge fields E( x, t) = ρ( x, t), B( x, t) = 0, B( x, t) = A( x, t) Quantum Electrodynamics (QED) for perturbative treatment [ ] E i ( x, t) = i A i ( x, t), [E i, A j ] = iδ ij, E ρ Ψ[A] = 0

20 Different descriptions of dynamical Abelian gauge fields: Maxwell s classical electromagnetic gauge fields E( x, t) = ρ( x, t), B( x, t) = 0, B( x, t) = A( x, t) Quantum Electrodynamics (QED) for perturbative treatment [ ] E i ( x, t) = i A i ( x, t), [E i, A j ] = iδ ij, E ρ Ψ[A] = 0 Wilson s U(1) lattice gauge theory for classical simulation ( y ) U xy = exp ie d l A = exp(iϕ xy ) U(1), E xy = i, x ϕ xy [ ] [E xy, U xy ] = U xy, (E E x,x+î x î,x ) ρ Ψ[U] = 0 i

21 Different descriptions of dynamical Abelian gauge fields: Maxwell s classical electromagnetic gauge fields E( x, t) = ρ( x, t), B( x, t) = 0, B( x, t) = A( x, t) Quantum Electrodynamics (QED) for perturbative treatment [ ] E i ( x, t) = i A i ( x, t), [E i, A j ] = iδ ij, E ρ Ψ[A] = 0 Wilson s U(1) lattice gauge theory for classical simulation ( y ) U xy = exp ie d l A = exp(iϕ xy ) U(1), E xy = i, x ϕ xy [ ] [E xy, U xy ] = U xy, (E E x,x+î x î,x ) ρ Ψ[U] = 0 i U(1) quantum link models for quantum simulation U xy = S + xy, U xy = S xy, E xy = S 3 xy, [E xy, U xy ] = U xy, [E xy, U xy] = U xy, [U xy, U xy] = 2E xy

22 U(1) gauge fields from spins 1 2 U = S +, U = S, E = S 3. Gauss law x E x,i U x,i x + î Ring-exchange plaquette Hamiltonian H = J H = 0 D. Horn, Phys. Lett. B100 (1981) 149 P. Orland, D. Rohrlich, Nucl. Phys. B338 (1990) 647 S. Chandrasekharan, UJW, Nucl. Phys. B492 (1997) 455

23 Energy density of charge-anti-charge pair Q = ±2 (a) d 60 0 (b) (c) 0 60 c -1 (d) D. Banerjee, F.-J. Jiang, P. Widmer, UJW, JSTAT (2013) P12010.

24 String theory on a chip with superconducting circuits D. Marcos, P. Rabl, E. Rico, P. Zoller, Phys. Rev. Lett. 111 (2013) (2013). D. Marcos, P. Widmer, E. Rico, M. Hafezi, P. Rabl, UJW, P. Zoller, arxiv:

25 P-state excited Rydberg atoms in an optical lattice A. G. Glaetzle, M. Dalmonte, R. Nath, I. Rousochatzakis, R. Moessner, P. Zoller, arxiv:

26 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

27 Analog quantum simulator proposals H. P. Büchler, M. Hermele, S. D. Huber, M. P. A. Fisher, P. Zoller, Phys. Rev. Lett. 95 (2005) E. Zohar, B. Reznik, Phys. Rev. Lett. 107 (2011) E. Zohar, J. I. Cirac, B. Reznik, Phys. Rev. Lett. 109 (2012) ; Phys. Rev. Lett. 110 (2013) ; Phys. Rev. Lett. 110 (2013) D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, UJW, P. Zoller, Phys. Rev. Lett. 109 (2012) D. Banerjee, M. Bögli, M. Dalmonte, E. Rico, P. Stebler, UJW, P. Zoller, Phys. Rev. Lett. 110 (2013) Digital quantum simulator proposals M. Müller, I. Lesanovsky, H. Weimer, H. P. Büchler, P. Zoller, Phys. Rev. Lett. 102 (2009) ; Nat. Phys. 6 (2010) 382. L. Tagliacozzo, A. Celi, P. Orland, M. Lewenstein, Nature Communications 4 (2013) L. Tagliacozzo, A. Celi, A. Zamora, M. Lewenstein, Ann. Phys. 330 (2013) 160. Review on quantum simulators for lattice gauge theories UJW, Annalen der Physik 525 (2013) 777, arxiv:

28 Hamiltonian for staggered fermions and U(1) quantum links H = t [ ] ψ xu x,x+1 ψ x+1 + h.c. + m ( 1) x ψ xψ x + g 2 2 x x x U x,x+1 = b x b x+1, E x,x+1 = 1 ) (b x+1 2 b x+1 b xb x E 2 x,x+1

29 Hamiltonian for staggered fermions and U(1) quantum links H = t [ ] ψ xu x,x+1 ψ x+1 + h.c. + m ( 1) x ψ xψ x + g 2 2 x x x U x,x+1 = b x b x+1, E x,x+1 = 1 ) (b x+1 2 b x+1 b xb x Optical lattice with Bose-Fermi mixture of ultra-cold atoms a) b 1 F b 2 U t F t B E 2 x,x+1 b) 2(U + m) x 2U t F 2U b 1 F 2 U + g2 4 t 2 B 2U t B 2U b 2

Bose-Fermi mixture of ultra-cold atoms a) b 1 F b 2 U t

30 Hamiltonian for staggered fermions and U(1) quantum links H = t [ ] ψ xu x,x+1 ψ x+1 + h.c. + m ( 1) x ψ xψ x + g 2 2 x x x U x,x+1 = b x b x+1, E x,x+1 = 1 ) (b x+1 2 b x+1 b xb x Optical lattice with Bose-Fermi mixture of ultra-cold atoms a) b 1 F b 2 U t F t B E 2 x,x+1 b) 2(U + m) x 2U t F 2U b 1 F 2 U + g2 4 t 2 B 2U t B 2U b 2

31 Quantum simulation of the real-time evolution of string breaking x E c) t τ t F /U t τ D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, UJW, P. Zoller, PRL 109 (2012) F. Hebenstreit, J. Berges, D. Gelfand, PRD 87 (2013) S. Kühn, J. I. Cirac, M. Banuls, PRA 90 (2014) T. Pichler, M. Dalmonte, E. Rico, P. Zoller, S. Montagnero, arxiv: V. Kasper, F. Hebenstreit, M. Oberthaler, J. Berges, arxiv:

32 U(N) quantum link operators U ij = S ij ij 1 +is 2, Uij = S ij 1 is ij 2, i, j {1, 2,..., N}, [Uij, (U ) kl ] 0 SU(N) L SU(N) R gauge transformations of a quantum link [L a, L b ] = if abc L c, [R a, R b ] = if abc R c, a, b, c {1, 2,..., N 2 1} [L a, R b ] = [L a, E] = [R a, E] = 0 Infinitesimal gauge transformations of a quantum link [L a, U] = λ a U, [R a, U] = Uλ a, [E, U] = U Algebraic structures of different quantum link models U(N) : U ij, L a, R a, E, 2N 2 +2(N 2 1)+1 = 4N 2 1 SU(2N) generators SO(N) : O ij, L a, R a, N 2 N(N 1) +2 = N(2N 1) SO(2N) generators 2 Sp(N) : U ij, L a, R a, 4N 2 +2N(2N+1) = 2N(4N+1) Sp(2N) generators R. Brower, S. Chandrasekharan, UJW, Phys. Rev. D60 (1999)

33 Low-energy effective action of a quantum link model β S[G µ ] = dx 5 d 4 x 1 ( 2e 2 Tr G µν G µν + 1 ) c 2 Tr 5G µ 5 G µ, G 5 = 0 0 undergoes dimensional reduction from to 4 dimensions S[G µ ] d 4 1 x 2g 2 Tr G 1 µνg µν, g 2 = β ( e 2, 1 24π 2 ) m exp β 11Ne 2

34 SU(3) quantum link operator in terms of fermionic rishons U ij xy = c i xc j y, i, j {1, 2, 3} Ring-exchange Hamiltonian as a rishon-abacus

35 Optical lattice with ultra-cold alkaline-earth atoms ( 87 Sr or 173 Yb) with color encoded in nuclear spin a) i x U ij x,x+1 b) t x 1 c i x, x x +1 i x c i x,+ t +t t c) e) ( t U "i #i "i #i 3/2 1/2 1/2 3/2 ( t U d) t D. Banerjee, M. Bögli, M. Dalmonte, E. Rico, P. Stebler, UJW, P. Zoller, Phys. Rev. Lett. 110 (2013)

36 Expansion of a fireball mimicking a hot quark-gluon plasma 1 τ = 0t 1 τ = 1t 1 τ = 10t 1 ( ψψ)x x

37 How to reach the continuum limit? Ladder of SU(N) quantum spins [T a x, T b y ] = iδ xy f abc T c x embodied with alkaline-earth atoms. H = J [Tx a T a x+ˆ1 + T x a T a x+ˆ2 ] J x A x B [T a x T a x+ˆ1 + T a x T a x+ˆ2 ] L y x L Very large correlation length ξ exp(4πl ρ s /cn) L. Reduction to the (1 + 1)-d CP(N 1) model at θ = nπ. β L { [ 1 S[P] = dt dx Tr g 2 x P x P + 1 ] } c 2 tp t P np x P t P 0 0 M. Dalmonte, C. Kraus, C. Laflamme, E. Rico, UJW, P. Zoller, in preparation

38 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

39 Nuclear Physics from SU(3) QCD Quarks Baryon Nucleus Gluon

40 Nuclear Physics from SU(3) QCD Quarks Baryon Nucleus Gluon Nuclear Physics in an SO(3) lattice gauge theory? SO(3) "Baryon" SO(3) "Nucleus"

41 1-d SO(3) quantum link model with adjoint triplet-fermions H = t [ ] ψx i O ij x,x+1 ψj x+1 + h.c. + m ( 1) x ψx i ψx i x x SO(3) quantum links O ij x,x+1 = σi x,l σj x+1,r Encoding manifestly gauge invariant states obeying Gauss law

42 Restoration of chiral symmetry at baryon density n B E with constant Baryon density n B 1 E = E B+ - E B n B = 0 n B = 1/2 n B = 1 1e M. Dalmonte, E. Rico, D. Banerjee, M. Bögli, P. Stebler, UJW, P. Zoller, in preparation L

43 Implementation with magnetic atoms (e.g. Cr), whose dipolar interactions allow spin-spin interactions without superexchange A. de Paz, A. Sharma, A. Chotia, E. Marechal, J. H. Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, and B. Laburthe-Tolra, Phys. Rev. Lett. 111 (2013)

44 Outline Motivation from Nuclear and Particle Physics Classical and Quantum Simulations of Quantum Spin Systems Quantum Link Formulation of Lattice Gauge Theories Atomic Quantum Simulators for Lattice Gauge Theories Nuclear Physics from an SO(3) Gauge Model via Encoding Conclusions

45 Conclusions Quantum link models provide an alternative formulation of lattice gauge theory with a finite-dimensional Hilbert space per link, which allows implementations with ultra-cold atoms in optical lattices.

46 Conclusions Quantum link models provide an alternative formulation of lattice gauge theory with a finite-dimensional Hilbert space per link, which allows implementations with ultra-cold atoms in optical lattices. Quantum link models can be formulated with manifestly gauge invariant degrees of freedom that characterize the realization of the Gauss law. Encoding these degrees of freedom, e.g. in magnetic atoms with dipolar interactions, offers a new robust way to protect gauge invariance.

47 Conclusions Quantum link models provide an alternative formulation of lattice gauge theory with a finite-dimensional Hilbert space per link, which allows implementations with ultra-cold atoms in optical lattices. Quantum link models can be formulated with manifestly gauge invariant degrees of freedom that characterize the realization of the Gauss law. Encoding these degrees of freedom, e.g. in magnetic atoms with dipolar interactions, offers a new robust way to protect gauge invariance. Quantum simulator constructions have already been presented for the U(1) quantum link model as well as for U(N) and SU(N) quantum link models with fermionic matter, using ultra-cold Bose-Fermi mixtures or alkaline-earth atoms.

48 Conclusions Quantum link models provide an alternative formulation of lattice gauge theory with a finite-dimensional Hilbert space per link, which allows implementations with ultra-cold atoms in optical lattices. Quantum link models can be formulated with manifestly gauge invariant degrees of freedom that characterize the realization of the Gauss law. Encoding these degrees of freedom, e.g. in magnetic atoms with dipolar interactions, offers a new robust way to protect gauge invariance. Quantum simulator constructions have already been presented for the U(1) quantum link model as well as for U(N) and SU(N) quantum link models with fermionic matter, using ultra-cold Bose-Fermi mixtures or alkaline-earth atoms. This allows the quantum simulation of the real-time evolution of string breaking as well as the quantum simulation of nuclear physics and dense quark matter, at least in a qualitative SO(3) toy model for QCD.

49 Conclusions Quantum link models provide an alternative formulation of lattice gauge theory with a finite-dimensional Hilbert space per link, which allows implementations with ultra-cold atoms in optical lattices. Quantum link models can be formulated with manifestly gauge invariant degrees of freedom that characterize the realization of the Gauss law. Encoding these degrees of freedom, e.g. in magnetic atoms with dipolar interactions, offers a new robust way to protect gauge invariance. Quantum simulator constructions have already been presented for the U(1) quantum link model as well as for U(N) and SU(N) quantum link models with fermionic matter, using ultra-cold Bose-Fermi mixtures or alkaline-earth atoms. This allows the quantum simulation of the real-time evolution of string breaking as well as the quantum simulation of nuclear physics and dense quark matter, at least in a qualitative SO(3) toy model for QCD. Accessible effects may include chiral symmetry restoration, baryon superfluidity, or color superconductivity at high baryon density, as well as the quantum simulation of nuclear collisions.

50 Conclusions Quantum link models provide an alternative formulation of lattice gauge theory with a finite-dimensional Hilbert space per link, which allows implementations with ultra-cold atoms in optical lattices. Quantum link models can be formulated with manifestly gauge invariant degrees of freedom that characterize the realization of the Gauss law. Encoding these degrees of freedom, e.g. in magnetic atoms with dipolar interactions, offers a new robust way to protect gauge invariance. Quantum simulator constructions have already been presented for the U(1) quantum link model as well as for U(N) and SU(N) quantum link models with fermionic matter, using ultra-cold Bose-Fermi mixtures or alkaline-earth atoms. This allows the quantum simulation of the real-time evolution of string breaking as well as the quantum simulation of nuclear physics and dense quark matter, at least in a qualitative SO(3) toy model for QCD. Accessible effects may include chiral symmetry restoration, baryon superfluidity, or color superconductivity at high baryon density, as well as the quantum simulation of nuclear collisions. The path towards quantum simulation of QCD will be a long one. However, with a lot of interesting physics along the way.

Atomic Quantum Simulation of Abelian and non-abelian Gauge Theories

Atomic Quantum Simulation of Abelian and non-abelian Gauge Theories Uwe-Jens Wiese Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Bern University CERN, January 29, 2014