Quantum simulation of an extra dimension
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- Jonah Booker
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1 Quantum simulation of an extra dimension Alessio Celi based on PRL 108, (2012), with O. Boada, J.I. Latorre, M. Lewenstein, Quantum Technologies Conference III QTC III, Warzsawa, 14/09/2012 p. 1/14
2 Outline: Idea of Quantum Simulator (Feynman) QTC III, Warzsawa, 14/09/2012 p. 2/14
3 Outline: Idea of Quantum Simulator (Feynman) Cold Atoms QTC III, Warzsawa, 14/09/2012 p. 2/14
4 Outline: Idea of Quantum Simulator (Feynman) Cold Atoms Optical lattice: Hubbard model with internal degrees of freedom QTC III, Warzsawa, 14/09/2012 p. 2/14
5 Outline: Idea of Quantum Simulator (Feynman) Cold Atoms Optical lattice: Hubbard model with internal degrees of freedom Old Ideas (and applications): Synthetic gauge field: Jaksch and Zoller proposal and extensions QTC III, Warzsawa, 14/09/2012 p. 2/14
6 Outline: Idea of Quantum Simulator (Feynman) Cold Atoms Optical lattice: Hubbard model with internal degrees of freedom Old Ideas (and applications): Synthetic gauge field: Jaksch and Zoller proposal and extensions New use: Models with extra-dimension (in particular 4D) QTC III, Warzsawa, 14/09/2012 p. 2/14
7 Outline: Idea of Quantum Simulator (Feynman) Cold Atoms Optical lattice: Hubbard model with internal degrees of freedom Old Ideas (and applications): Synthetic gauge field: Jaksch and Zoller proposal and extensions New use: Models with extra-dimension (in particular 4D) Models with twisted boundary conditions (in progress) QTC III, Warzsawa, 14/09/2012 p. 2/14
8 Outline: Idea of Quantum Simulator (Feynman) Cold Atoms Optical lattice: Hubbard model with internal degrees of freedom Old Ideas (and applications): Synthetic gauge field: Jaksch and Zoller proposal and extensions New use: Models with extra-dimension (in particular 4D) Models with twisted boundary conditions (in progress) Prospects QTC III, Warzsawa, 14/09/2012 p. 2/14
9 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V QTC III, Warzsawa, 14/09/2012 p. 3/14
10 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) QTC III, Warzsawa, 14/09/2012 p. 3/14
11 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) New field: Quantum computation QTC III, Warzsawa, 14/09/2012 p. 3/14
12 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) New field: Quantum computation Universal quantum computer still far away QTC III, Warzsawa, 14/09/2012 p. 3/14
13 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) New field: Quantum computation Universal quantum computer still far away/dedicated quantum simulator possible QTC III, Warzsawa, 14/09/2012 p. 3/14
14 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) New field: Quantum computation Universal quantum computer still far away/dedicated quantum simulator possible Requirements: reproduces H eff (λ i ) parameters λ i tunable QTC III, Warzsawa, 14/09/2012 p. 3/14
15 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) New field: Quantum computation Universal quantum computer still far away/dedicated quantum simulator possible Requirements: reproduces H eff (λ i ) Analogic vs Digital: reproduces e ih eff(λ i )t parameters λ i tunable QTC III, Warzsawa, 14/09/2012 p. 3/14
16 Why a Quantum Simulator? (after discretization) In a quantum system dim. H grows exponentially with V Feynman (1983): Classical computer simulation takes exponential time Hypothetical quantum computer does not ( polynomial) New field: Quantum computation Universal quantum computer still far away/dedicated quantum simulator possible Requirements: reproduces H eff (λ i ) Analogic vs Digital: reproduces e ih eff(λ i )t parameters λ i tunable Good candidate: Cold atoms QTC III, Warzsawa, 14/09/2012 p. 3/14
17 Untractable vs Unaccessible Two lines of interest in Quantum simulation QTC III, Warzsawa, 14/09/2012 p. 4/14
18 Untractable vs Unaccessible Two lines of interest in Quantum simulation Solution of non-calculable models QTC III, Warzsawa, 14/09/2012 p. 4/14
19 Untractable vs Unaccessible Two lines of interest in Quantum simulation Solution of non-calculable models Examples: Superconductivity (Non-abelian lattice) Gauge theory Non-equilibrium dynamics and time evolution... QTC III, Warzsawa, 14/09/2012 p. 4/14
20 Untractable vs Unaccessible Two lines of interest in Quantum simulation Solution of non-calculable models Examples: Superconductivity (Non-abelian lattice) Gauge theory Non-equilibrium dynamics and time evolution... Design of systems unobservable (or yet unobserved) QTC III, Warzsawa, 14/09/2012 p. 4/14
21 Untractable vs Unaccessible Two lines of interest in Quantum simulation Solution of non-calculable models Examples: Superconductivity (Non-abelian lattice) Gauge theory Non-equilibrium dynamics and time evolution... Design of systems unobservable (or yet unobserved) Examples: Zitterbewegung, Klein paradox Neutrino oscillations (direct measurement) Dirac fermion in artificially curved spacetime... QTC III, Warzsawa, 14/09/2012 p. 4/14
22 Untractable vs Unaccessible Two lines of interest in Quantum simulation Solution of non-calculable models Examples: Superconductivity (Non-abelian lattice) Gauge theory Non-equilibrium dynamics and time evolution... Design of systems unobservable (or yet unobserved) Examples: Zitterbewegung, Klein paradox Neutrino oscillations (direct measurement) Dirac fermion in artificially curved spacetime... Simulation of models with four spatial dimensions QTC III, Warzsawa, 14/09/2012 p. 4/14
23 Untractable vs Unaccessible Two lines of interest in Quantum simulation Solution of non-calculable models Examples: Superconductivity (Non-abelian lattice) Gauge theory Non-equilibrium dynamics and time evolution... Design of systems unobservable (or yet unobserved) Examples: Zitterbewegung, Klein paradox Neutrino oscillations (direct measurement) Dirac fermion in artificially curved spacetime... Simulation of models with four spatial dimensions Note that once interactions are included the above problems become not calculable QTC III, Warzsawa, 14/09/2012 p. 4/14
24 Observations & Ingredients for the simulation All the example mentioned can be conveniently formulated on the lattice QTC III, Warzsawa, 14/09/2012 p. 5/14
25 Observations & Ingredients for the simulation All the example mentioned can be conveniently formulated on the lattice Natural set-up: Optical lattices QTC III, Warzsawa, 14/09/2012 p. 5/14
26 Observations & Ingredients for the simulation All the example mentioned can be conveniently formulated on the lattice Natural set-up: Optical lattices The Hamiltonian to be simulated reduces to extended Hubbard model with internal degrees of freedom QTC III, Warzsawa, 14/09/2012 p. 5/14
27 Observations & Ingredients for the simulation All the example mentioned can be conveniently formulated on the lattice Natural set-up: Optical lattices The Hamiltonian to be simulated reduces to extended Hubbard model with internal degrees of freedom Natural ingredientes: Hyperfine states, Superlattice QTC III, Warzsawa, 14/09/2012 p. 5/14
28 Observations & Ingredients for the simulation All the example mentioned can be conveniently formulated on the lattice Natural set-up: Optical lattices The Hamiltonian to be simulated reduces to extended Hubbard model with internal degrees of freedom Natural ingredientes: Hyperfine states, Superlattice (Historical) Prototype: Simulation of Synthetic gauge field in optical lattices QTC III, Warzsawa, 14/09/2012 p. 5/14
29 Observations & Ingredients for the simulation All the example mentioned can be conveniently formulated on the lattice Natural set-up: Optical lattices The Hamiltonian to be simulated reduces to extended Hubbard model with internal degrees of freedom Natural ingredientes: Hyperfine states, Superlattice (Historical) Prototype: Simulation of Synthetic gauge field in optical lattices We borrow some ideas from there QTC III, Warzsawa, 14/09/2012 p. 5/14
30 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important QTC III, Warzsawa, 14/09/2012 p. 6/14
31 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed QTC III, Warzsawa, 14/09/2012 p. 6/14
32 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics QTC III, Warzsawa, 14/09/2012 p. 6/14
33 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics Problem: atoms are neutral QTC III, Warzsawa, 14/09/2012 p. 6/14
34 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics Problem: atoms are neutral Old Idea: rotation constant magnetic field (Larmor theorem) QTC III, Warzsawa, 14/09/2012 p. 6/14
35 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics Problem: atoms are neutral Old Idea: rotation constant magnetic field (Larmor theorem) magnetic flux limited QTC III, Warzsawa, 14/09/2012 p. 6/14
36 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics Problem: atoms are neutral Old Idea: rotation constant magnetic field (Larmor theorem) magnetic flux limited Jaksch and Zoller,2003 (theory): artificial magnetic field with Raman lasers in OL QTC III, Warzsawa, 14/09/2012 p. 6/14
37 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics Problem: atoms are neutral Old Idea: rotation constant magnetic field (Larmor theorem) magnetic flux limited Jaksch and Zoller,2003 (theory): artificial magnetic field with Raman lasers in OL access the Hofstadter butterfly s physics QTC III, Warzsawa, 14/09/2012 p. 6/14
38 Synthetic (external) gauge field Motivation: Interesting strong coupling phenomena where the charge of electron is important Examples: Quantum Hall effect: a constant strong magnetic field needed More ambitious: Chromodynamics Problem: atoms are neutral Old Idea: rotation constant magnetic field (Larmor theorem) magnetic flux limited Jaksch and Zoller,2003 (theory): artificial magnetic field with Raman lasers in OL access the Hofstadter butterfly s physics Quickly developing research area QTC III, Warzsawa, 14/09/2012 p. 6/14
39 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. QTC III, Warzsawa, 14/09/2012 p. 7/14
40 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. Hofstadter,1976: coupling to an external background field, A vector potential QTC III, Warzsawa, 14/09/2012 p. 7/14
41 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. Hofstadter,1976: coupling to an external background field, A vector potential Peierls substitution: pseudo-momentum k k A QTC III, Warzsawa, 14/09/2012 p. 7/14
42 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. Hofstadter,1976: coupling to an external background field, A vector potential Peierls substitution: pseudo-momentum k k A Phase changing hopping ( Wilson line) H = J x m,n= e ia x(m,n) a m+1,n a m,n J y m,n= e ia y(m,n) a m,n+1 a m,n +h.c. QTC III, Warzsawa, 14/09/2012 p. 7/14
43 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. Hofstadter,1976: coupling to an external background field, A vector potential Peierls substitution: pseudo-momentum k k A Phase changing hopping ( Wilson line) H = J x m,n= e ia x(m,n) a m+1,n a m,n J y m,n= e ia y(m,n) a m,n+1 a m,n +h.c. Landau gauge Constant magnetic field B = (0,0,B) A = (0, Bx, 0), QTC III, Warzsawa, 14/09/2012 p. 7/14
44 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. Hofstadter,1976: coupling to an external background field, A vector potential Peierls substitution: pseudo-momentum k k A Phase changing hopping ( Wilson line) H = J x m,n= e ia x(m,n) a m+1,n a m,n J y m,n= e ia y(m,n) a m,n+1 a m,n +h.c. Landau gauge Constant magnetic field B = (0,0,B) A = (0, Bx, 0), QTC III, Warzsawa, 14/09/2012 p. 7/14
45 Gauging the Hubbard model Example: Electron (without spin) in a 2d crystal, Hubbard model (small filling) H = J x m,n= a m+1,n a m,n J y m,n= a m,n+1 a m,n +h.c. Hofstadter,1976: coupling to an external background field, A vector potential Peierls substitution: pseudo-momentum k k A Phase changing hopping ( Wilson line) H = J x m,n= e ia x(m,n) a m+1,n a m,n J y m,n= e ia y(m,n) a m,n+1 a m,n +h.c. Landau gauge Constant magnetic field B = (0,0,B) A = (0, Bx, 0), Generalization to non abelian theory in D dim.: a m,n a σ r, r R D, ex. SU(N), σ = 1,...N A µ A I µt I, T I gauge g. generators, hopping phase hopping matrix QTC III, Warzsawa, 14/09/2012 p. 7/14
46 Jaksch-Zoller (JZ) proposal Two ingredients QTC III, Warzsawa, 14/09/2012 p. 8/14
47 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e QTC III, Warzsawa, 14/09/2012 p. 8/14
48 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n QTC III, Warzsawa, 14/09/2012 p. 8/14
49 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n, V 0y V 0x J 0y suppressed QTC III, Warzsawa, 14/09/2012 p. 8/14
50 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n, V 0y V 0x J 0y suppressed Stimulated Rabi osc. between g and e by two running Raman lasers: Ω(x) = Ω 0 e i(k e k g ) x QTC III, Warzsawa, 14/09/2012 p. 8/14
51 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n, V 0y V 0x J 0y suppressed Stimulated Rabi osc. between g and e by two running Raman lasers: Ω(x) = Ω 0 e i(k e k g ) x, perturbation of OL (same Wannier), k g k e = 2π λ QTC III, Warzsawa, 14/09/2012 p. 8/14
52 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n, V 0y V 0x J 0y suppressed Stimulated Rabi osc. between g and e by two running Raman lasers: Ω(x) = Ω 0 e i(k e k g ) x, perturbation of OL (same Wannier), k g k e = 2π λ H = J 0x Final result a m+1,n a m,n + J y e iay(m,n) a m,n+1 a m,n +h.c., m,n= m,n= QTC III, Warzsawa, 14/09/2012 p. 8/14
53 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n, V 0y V 0x J 0y suppressed Stimulated Rabi osc. between g and e by two running Raman lasers: Ω(x) = Ω 0 e i(k e k g ) x, perturbation of OL (same Wannier), k g k e = 2π λ H = J 0x m,n= Final result a m+1,n a m,n + J y e iay(m,n) a m,n+1 a m,n +h.c., m,n= Average over Wannier J y e ia y(m,n) = 2 d 2 xw (x x m,n )Ωw(x x m,n+1 ). QTC III, Warzsawa, 14/09/2012 p. 8/14
54 Jaksch-Zoller (JZ) proposal Two ingredients Superposition of two 2D lattices loaded with two different hyperfine states g and e Off-set of λ/4 in y direction, x m = λ/2m, y n = λ/4n, V 0y V 0x J 0y suppressed Stimulated Rabi osc. between g and e by two running Raman lasers: Ω(x) = Ω 0 e i(k e k g ) x, perturbation of OL (same Wannier), k g k e = 2π λ H = J 0x m,n= Final result a m+1,n a m,n + J y e iay(m,n) a m,n+1 a m,n +h.c., m,n= Average over Wannier J y e ia y(m,n) = 2 d 2 xw (x x m,n )Ωw(x x m,n+1 ). For k e k g //ˆx A y (m,n) = 2πΦm, Constant magnetic field in Landau gauge QTC III, Warzsawa, 14/09/2012 p. 8/14
55 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) QTC III, Warzsawa, 14/09/2012 p. 9/14
56 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: QTC III, Warzsawa, 14/09/2012 p. 9/14
57 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) QTC III, Warzsawa, 14/09/2012 p. 9/14
58 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, QTC III, Warzsawa, 14/09/2012 p. 9/14
59 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, Superlattices, etc EX. Toolbox... by Mazza et al., New J. Phys. 14 (2012) QTC III, Warzsawa, 14/09/2012 p. 9/14
60 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, Superlattices, etc EX. Toolbox... by Mazza et al., New J. Phys. 14 (2012) Shaking cf talk K. Sacha & K.Sengstock QTC III, Warzsawa, 14/09/2012 p. 9/14
61 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, Superlattices, etc EX. Toolbox... by Mazza et al., New J. Phys. 14 (2012) Shaking cf talk K. Sacha & K.Sengstock Osterloh et al, Ruseckas et al, 2005 Hyperfine states as internal gauge d.o.f non abelian interactions QTC III, Warzsawa, 14/09/2012 p. 9/14
62 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, Superlattices, etc EX. Toolbox... by Mazza et al., New J. Phys. 14 (2012) Shaking cf talk K. Sacha & K.Sengstock Osterloh et al, Ruseckas et al, 2005 Hyperfine states as internal gauge d.o.f non abelian interactions Ex: SU(2), hopping phase 2 2 unitary hopping matrices QTC III, Warzsawa, 14/09/2012 p. 9/14
63 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, Superlattices, etc EX. Toolbox... by Mazza et al., New J. Phys. 14 (2012) Shaking very recent Hamburg-Icfo-Dresden non-abelian PRL 110, xxxxxx (2012) Osterloh et al, Ruseckas et al, 2005 Hyperfine states as internal gauge d.o.f non abelian interactions Ex: SU(2), hopping phase 2 2 unitary hopping matrices QTC III, Warzsawa, 14/09/2012 p. 9/14
64 Comments... JZ proposal: experiment PRL (2012) (I. Bloch group) Many more proposals with or without lattice: Rotation (experiments since 2000) Dark state (Berry phase), first experiments NIST, Superlattices, etc EX. Toolbox... by Mazza et al., New J. Phys. 14 (2012) Shaking very recent Hamburg-Icfo-Dresden non-abelian PRL 110, xxxxxx (2012) Osterloh et al, Ruseckas et al, 2005 Hyperfine states as internal gauge d.o.f non abelian interactions Ex: SU(2), hopping phase 2 2 unitary hopping matrices Observations: classical background gauge field configuration (no dynamics) always gauge-fixed Hamiltonian relativistic matter & extradimension simulations require synthetic gauge field QTC III, Warzsawa, 14/09/2012 p. 9/14
65 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential QTC III, Warzsawa, 14/09/2012 p. 10/14
66 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential and > 3D? QTC III, Warzsawa, 14/09/2012 p. 10/14
67 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential and > 3D? In a lattice Dimensionality Connectivity QTC III, Warzsawa, 14/09/2012 p. 10/14
68 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential and > 3D? In a lattice Dimensionality Connectivity Basic idea: just kinetic term in D+1 hypercubic lattice H = J q D+1 j=1 a q+u j a q +H.c., QTC III, Warzsawa, 14/09/2012 p. 10/14
69 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential and > 3D? In a lattice Dimensionality Connectivity Basic idea: just kinetic term in D+1 hypercubic lattice H = J q D+1 j=1 a q+u j a q +H.c., q = (r,σ), D+1-space in D-layers H = J r,σ ( D j=1 a (σ) r+u j a (σ) r +a (σ+1) r ) a r (σ) +H.c., QTC III, Warzsawa, 14/09/2012 p. 10/14
70 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential and > 3D? In a lattice Dimensionality Connectivity Basic idea: just kinetic term in D+1 hypercubic lattice H = J q D+1 j=1 a q+u j a q +H.c., q = (r,σ), D+1-space in D-layers H = J r,σ ( D j=1 a (σ) r+u j a (σ) r +a (σ+1) r ) a r (σ) +H.c., a r (σ),a r (σ) as Fock generators of different species on same site, extra-dim internal d.o.f Hopping in extra-dim transmutation between contiguous species, QTC III, Warzsawa, 14/09/2012 p. 10/14
71 Modelling extra-dimensions (Boada,AC,Latorre,Lewenstein, PRL 2012) (during the workshop) in OL 3D Hubbard model 1D,2D by tuning optical potential and > 3D? In a lattice Dimensionality Connectivity Basic idea: just kinetic term in D+1 hypercubic lattice H = J q D+1 j=1 a q+u j a q +H.c., q = (r,σ), D+1-space in D-layers H = J r,σ ( D j=1 a (σ) r+u j a (σ) r +a (σ+1) r ) a r (σ) +H.c., a r (σ),a r (σ) as Fock generators of different species on same site, extra-dim internal d.o.f Hopping in extra-dim transmutation between contiguous species, 2 ways: State-dependent lattice (cf JZ proposal) On-site dressed lattice (cf relativistic fermions Toolbox) QTC III, Warzsawa, 14/09/2012 p. 10/14
72 Extra-dimension implementation & detection State-dependent & On-site dressed, common features Open or periodic boundary cond. in the extra-dim Compactification Complex hopping on the S 1 (p.b.c.) Flux compactification Interactions (on-site and long range) QTC III, Warzsawa, 14/09/2012 p. 11/14
73 Extra-dimension implementation & detection State-dependent & On-site dressed, common features Open or periodic boundary cond. in the extra-dim Compactification Complex hopping on the S 1 (p.b.c.) Flux compactification Interactions (on-site and long range) State-dependent & On-site dressed, specific features QTC III, Warzsawa, 14/09/2012 p. 11/14
74 Extra-dimension implementation & detection State-dependent & On-site dressed, common features Open or periodic boundary cond. in the extra-dim Compactification Complex hopping on the S 1 (p.b.c.) Flux compactification Interactions (on-site and long range) State-dependent & On-site dressed, specific features State-dependent lattice: easy realization Example: 87 Rb, F = 1, F = 2 Bivolume N = 2 layers d QTC III, Warzsawa, 14/09/2012 p. 11/14
75 Extra-dimension implementation & detection State-dependent & On-site dressed, common features Open or periodic boundary cond. in the extra-dim Compactification Complex hopping on the S 1 (p.b.c.) Flux compactification Interactions (on-site and long range) State-dependent & On-site dressed, specific features State-dependent lattice: easy realization Example: 87 Rb, F = 1, F = 2 Bivolume N = 2 layers On-site dressed lattice: up to N = 10, (alkali-earth SU(N) inv. interaction) Example: 87 Sr, 40 K, F = 9 2 (fermions) QTC III, Warzsawa, 14/09/2012 p. 11/14
76 Extra-dimension implementation & detection State-dependent & On-site dressed, common features Open or periodic boundary cond. in the extra-dim Compactification Complex hopping on the S 1 (p.b.c.) Flux compactification Interactions (on-site and long range) State-dependent & On-site dressed, specific features State-dependent lattice: easy realization Example: 87 Rb, F = 1, F = 2 Bivolume N = 2 layers On-site dressed lattice: up to N = 10, (alkali-earth SU(N) inv. interaction) Example: 87 Sr, 40 K, F = 9 2 (fermions),? 167 Er, F = 19 2, N = 20? QTC III, Warzsawa, 14/09/2012 p. 11/14
77 Extra-dimension implementation & detection State-dependent & On-site dressed, common features Open or periodic boundary cond. in the extra-dim Compactification Complex hopping on the S 1 (p.b.c.) Flux compactification Interactions (on-site and long range) State-dependent & On-site dressed, specific features State-dependent lattice: easy realization Example: 87 Rb, F = 1, F = 2 Bivolume N = 2 layers On-site dressed lattice: up to N = 10, (alkali-earth SU(N) inv. interaction) Example: 87 Sr, 40 K, F = 9 2 (fermions),? 167 Er, F = 19 2, N = 20? Observables: dimensionality dependence Single-particle, scaling properties, Example: density of states Many-body, phase diagram, Example: MI-Superfluid conclusion QTC III, Warzsawa, 14/09/2012 p. 11/14
78 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible QTC III, Warzsawa, 14/09/2012 p. 12/14
79 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC QTC III, Warzsawa, 14/09/2012 p. 12/14
80 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? QTC III, Warzsawa, 14/09/2012 p. 12/14
81 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle QTC III, Warzsawa, 14/09/2012 p. 12/14
82 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle Simplest way: construct the full lattice with internal states QTC III, Warzsawa, 14/09/2012 p. 12/14
83 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle Simplest way: construct the full lattice with internal states Limitation: dimension of the lattice Advantage: many copies of it QTC III, Warzsawa, 14/09/2012 p. 12/14
84 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle Simplest way: construct the full lattice with internal states Limitation: dimension of the lattice Advantage: many copies of it Alternative: mixing of internal and spatial d.o.f Idea: Two internal states+single site addressing Scalable circle QTC III, Warzsawa, 14/09/2012 p. 12/14
85 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle Simplest way: construct the full lattice with internal states Limitation: dimension of the lattice Advantage: many copies of it Alternative: mixing of internal and spatial d.o.f Idea: Two internal states+single site addressing Scalable circle Four internal states+single site addressing Scalable Möbius strip QTC III, Warzsawa, 14/09/2012 p. 12/14
86 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle Simplest way: construct the full lattice with internal states Limitation: dimension of the lattice Advantage: many copies of it Alternative: mixing of internal and spatial d.o.f Idea: Two internal states+single site addressing Scalable circle Four internal states+single site addressing Scalable Möbius strip Interest: helicity modulus and stiffness in Heisenberg [Fisher 71], [Rossini, 11] QTC III, Warzsawa, 14/09/2012 p. 12/14
87 Exotic Boundary Conditions same people + J.Rodriguez-Laguna Usual optical lattices: only Open Boundary Condition (BC) are possible In lattices with compact extradimension or Bessel lattices: Periodic BC Can we do more? For instance, Twisted BC Möbius strip or Klein bottle Simplest way: construct the full lattice with internal states Limitation: dimension of the lattice Advantage: many copies of it Alternative: mixing of internal and spatial d.o.f Idea: Two internal states+single site addressing Scalable circle Four internal states+single site addressing Scalable Möbius strip Interest: helicity modulus and stiffness in Heisenberg [Fisher 71], [Rossini, 11] Additional ingredients: Generalized Twisted B.C. (with phases), Interactions QTC III, Warzsawa, 14/09/2012 p. 12/14
88 Prospects QTC III, Warzsawa, 14/09/2012 p. 13/14
89 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming QTC III, Warzsawa, 14/09/2012 p. 14/14
90 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming My plans, both in Uncalculable non-abelian link model (Gauge theory) QTC III, Warzsawa, 14/09/2012 p. 14/14
91 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming My plans, both in Uncalculable non-abelian link model (Gauge theory) and Unaccessible problems: QTC III, Warzsawa, 14/09/2012 p. 14/14
92 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming My plans, both in Uncalculable non-abelian link model (Gauge theory) and Unaccessible problems: Testing of Fulling-Unruh Thermalization theorem by simulation of artificial Dirac fermions in Rindler spacetime QTC III, Warzsawa, 14/09/2012 p. 14/14
93 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming My plans, both in Uncalculable non-abelian link model (Gauge theory) and Unaccessible problems: Testing of Fulling-Unruh Thermalization theorem by simulation of artificial Dirac fermions in Rindler spacetime Simulation of Gravitino relativistic spin 3 2 particle QTC III, Warzsawa, 14/09/2012 p. 14/14
94 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming My plans, both in Uncalculable non-abelian link model (Gauge theory) and Unaccessible problems: Testing of Fulling-Unruh Thermalization theorem by simulation of artificial Dirac fermions in Rindler spacetime Simulation of Gravitino relativistic spin 3 2 particle Extra-dimensions: exotic phases in Bosons (in progress) and Fermions QTC III, Warzsawa, 14/09/2012 p. 14/14
95 Prospects Very exciting time for Quantum simulation: Increasing number of theoretical proposals for QFT simulators in OL New experiments are coming My plans, both in Uncalculable non-abelian link model (Gauge theory) and Unaccessible problems: Testing of Fulling-Unruh Thermalization theorem by simulation of artificial Dirac fermions in Rindler spacetime Simulation of Gravitino relativistic spin 3 2 particle Extra-dimensions: exotic phases in Bosons (in progress) and Fermions Twisted Boundary Condition: robust effects in thermodynamics limits QTC III, Warzsawa, 14/09/2012 p. 14/14
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