Golden chain of strongly interacting Rydberg atoms Hosho Katsura (Gakushuin Univ.) Acknowledgment: Igor Lesanovsky (MUARC/Nottingham Univ. I. Lesanovsky & H.K., [arxiv:1204.0903]
Outline 1. Introduction Exotic particles in condensed matter systems What are Fibonacci anyons? 2. Golden chain of Fibonacci anyons Anyonic Hilbert space, F-matrix, & Braiding Hamiltonian & Low-energy spectra Relation to CFT & Integrable models 3. Strongly correlated Rydberg atoms What is special about Rydberg atoms Rydberg blockade Connection to Fibonacci anyons 4. Summary and future directions
Where you may have heard of exotic particles : Anyons Quasiparticles with fractional charges FQHE, 2d magnetic models (Kitaev model), 1. Abelian anyons: acquire a phase factor under braiding 2. Non-abelian anyons: unitary rotation in the ground space Fibonacci anyons: Read-Rezayi state, Levin-Wen model Majorana fermions Page 73, Prob. 3.4, (Peskin-Schroeder s textbook) Quantum Ising chain/2d Ising model Signature of Majorana end states Mourik et al., Science 336, 1003 (2012). Parafermions Fractionalized Majoranas Zn invariant integrable models (e.g. 3-state Potts model) Correlators of CFTs describing Read-Rezayi states Experimental Setup Clarke, Alicea, Shtengel [arxiv:1204.5479], M. Cheng [arxiv:1204.6084]
What are Fibonacci anyons? Quantum dimension Majorana case: quantum dimension: There are thus 2 N states for 2N Majorana zero modes. Fibonacci case: quantum dimension: Fusion rule: Precise meaning: There are two possible quantum states for two Fibonacci anyons. 1 has trivial statistics, has the braiding properties of a single Fibonacci anyon. 1 1 2 3 5 8 Fibonacci #: 3/2 1 1/2 0 1 3 1 3 8 1 2 5 1 2 5 1 2 3 4 5 6 N
Outline 1. Introduction Exotic particles in condensed matter systems What are Fibonacci anyons? 2. Golden chain of Fibonacci anyons Anyonic Hilbert space, F-matrix, & Braiding Hamiltonian & Low-energy spectra Relation to CFT & Integrable models 3. Strongly correlated Rydberg atoms What is special about Rydberg atoms Rydberg blockade Connection to Fibonacci anyons 4. Summary and future directions
Model What is the collective state of a set of Fibonacci anyons The ground state has a macroscopic degeneracy Soup of anyons Anyons approach each other and interact. The interactions will lift the degeneracy. 1d array of anyons (pinned) Anyonic analog of Heisenberg interaction A. Feiguin et al., PRL. 98, 160409 (2007). Penalize fusion outcome 1 or. Numerics: Finite-size scaling analysis, entanglement entropies, unique critical g.s., Low-energy physics = minimal model CFT (c<1) Mapping & exact solution Anyonic spin chain = Integrable RSOS model (Andrews-Baxter-Forrester)
Anyonic Hilbert space Fusion tree basis Orthonormal basis Hilbert space is spanned by all possible fusion paths. Fusion rules Example: Constraint: Is there any other basis where fusion of two anyons become explicit?
F-matrix and braiding S. Trebst et al., Prog. Theor. Phys. Suppl. 176, 384 (2008). F-matrix Consistency condition (pentagon relation) Fusion-tree basis: New basis: Braiding Consistency Braiding in the condition new basis (hexagon relation) = phase factor Braiding in the fusion-tree basis (complicated )
How to model interaction? Heisenberg interaction Anyonic equivalent triplet splitting / J splitting / J singlet Hamiltonian Projection to singlet - Need to discriminate the two two fusion outcomes and 1 Projection to Projection to triplet Projection to
Hamiltonian in the fusion-tree basis Projection to 1 : (Forget for the moment ) Hamiltonian : Numerical Results J>0: antiferromagnetic J<0: ferromagnetic DMRG & Exact diagonalization: A. Feiguin et al., PRL. 98, 160409 (2007). Critical ground state Finite-size gap : Entanglement entropy : CFT description! J>0 (antiferromagnetic case) J>0: c=7/10 J<0: c=4/5 Minimal model CFTs!
Low-energy spectra A. Feiguin et al., PRL. 98, 160409 (2007). J>0: c=7/10 Universality class: tricritical Ising model Finite-size scaling : Topologically protected criticality Primary fields: 0, 1/5, 6/5, 3, 3/40, 7/8 Scaling dimension Relevant perturbations: Prohibited by translation & topological symmetries.
Integrability of the golden chain Temperley-Lieb algebra Hamiltonian : Restricted-solid-on-solid (RSOS) model: Andrews-Baxter-Forrester, JSP 35, 193 (1984); D. A. Huse, PRB 30, 3908 (1984). (generalized) Hard-square model Off-critical generalization! Transfer matrix: Allowed local configurations Baxter s textbook Exclusion constraint Physics of Rydberg atoms!
Outline 1. Introduction Exotic particles in condensed matter systems What are Fibonacci anyons? 2. Golden chain of Fibonacci anyons Anyonic Hilbert space, F-matrix, & Braiding Hamiltonian & Low-energy spectra Relation to CFT & Integrable models 3. Strongly correlated Rydberg atoms What is special about Rydberg atoms Rydberg blockade Connection to Fibonacci anyons 4. Summary and future directions
radial density Atoms in highly excited states Back to high-school physics Rydberg formula (1888) : Bohr s model (1913) Z Hydrogen atom, alkali metal atoms (n n*) What are Rydberg atoms? Atoms in which one of the electrons is in the excited state with a very high principal quantum number (n). Atoms become very fat! Rubidium 39s 121 nm + - - r
What is special about Rydberg atoms? Scaling laws - Characteristic radius - Ionization potential - Level spacing - Characteristic velocity - Radiation lifetime Strongly exaggerated interaction Rydberg atoms in s-states interact via van-der-waals type interaction For typical n-values (n=40-80), the interaction is 10 orders of magnitude stronger than that of ground-state atoms. Traditional ultra cold atoms: interatomic interaction ~ a few khz Rydberg atoms: vdw interaction ~ tens of MHz ( R ~μm) Potential applications in quantum information & simulation of many-body physics: M. Saffman, T.G. Walker & K. Mølmer, RMP 82, 2313 (2010).
Rydberg blockade Spin-1/2 representation Single-atom Hamiltonian (Rotating-wave approximation) Laser Rabi frequency Detuning Laser Blockade effect D. Jaksch et al., PRL 85, 2208 (2000); M. Lukin et al., PRL 87, 037901 (2001) van-der-waals type int. blockade radius Entangled superatom + +
Model Setup - Atoms trapped in 1d optical lattice - Interaction decays quickly (~ r --6 ) Consider only up to n.n.n. interaction van-der-waals int. Hamiltonian Bring the blockade into play! (with appropriate lattice spacing) We never have adjacent excited states Effective Hamiltonian projection onto space w/o adjacent excitations - Blockade is now explicit in the Hamiltonian - We are working in a subspace in which n i n i+1 =0 Looks complicated Why should anyons care?
Connection to Fibonacci anyons Hilbert space (with exclusion constraint) Fusion rule (reminder): Golden chain Physical No adjacent 1s No adjacent Rydberg states Rydberg atom Fictitious Fictitious Physical Local (3-atom) Hamiltonian Anyons Rydberg atoms when
Practical implementation Parameter region Sign of Rydberg-Rydberg interaction Both positive and negative coeficient V are available. Negative V: R. Mukherjee et al., J. Phys. B. 44, 184010 (2011); R. Low et al., PRL 106, 170401 (2011). Effect of imperfections - Non-perfect blockade The blockade can work! Low-energy spectra - Longer-range interactions Negative V (AFM: c=7/10): critical g.s. Positive V (FM: c=4/5): critical g.s. Robust! Gapped
Measurement of anyonic observables - quantity of interest is e.g. fusion outcome of neighboring anyons - need to perform projective measurement - amounts to three independent single site measurements Projective measurement of a single atom E. Urban et al., Nature Phys. 5, 110 (2009). A. Gaetan et al., Nature Phys. 5, 115 (2009).
Summary Realization of golden chain 1. Rydberg blockade restricted Hilbert space 2. Anyonic degrees of freedom are encoded in multiple atoms. 3. Experimental simulation of the anyonic Heisenberg chain. 4. Measurement scheme for fusion outcomes Rydberg atom system Platform for the study of interacting anyons Future directions - Fibonacci anyononic J1-J2 chain (2-body & 3-body interactions) S. Trebst et al., PRL 101, 050401 (2008). Stable critical phases - two-leg ladders, higher- spin models, two-dimensional models,... - Other class of exotic systems with Rydberg atoms Zamolodchikov s Ising field thery (E 8 symmetry and boud states) cf) CMT realization: CoNb 2 O 6, Coldea et al., Science 327, 177 (2010).
Supplement: F-matrix & Braiding F-matrix (basis transformation) Braiding (in the fusion-tree basis) Supplement: Two-dimensional Rydberg gas - Twi-dimensional Rydberg gases and quantum hard-square model S. Ji, C. Ates & I. Lesanovsky, PRL 107, 060406 (201). Allowed local configurations
Supplement: ground state phase diagram The same model also describes trapped atoms in a tilted optical lattice. P. Fendley, K. Sengupta & S. Sachdev, PRB 69, 075106 (2000). - Rokhsar-Kivelson line Igor Lesanovsky, PRL 106, 025301 (2011); Igor Lesanovsky, PRL 108, 105301 (2012). RK - RK ground state - Exact 1st excited state is also obtained