Entanglement and Rydberg interacon Antoine Browaeys, Instut d’Opque, CNRS Outline 1. Basics of entanglement 2. Entanglement and 2‐atom gate using Rybderg blockade Outline 1. Basics of entanglement 2. Entanglement and 2‐atom gate using Rybderg blockade Goal: control Rydberg interacon between few (10 – 100) atoms (No ensemble average) Quantum state engineering Outline 1. Basics of entanglement 2. Entanglement and 2‐atom gate using Rybderg blockade What is entanglement? 2 systems A, B 2 d° of freedom 2 modes Superposion principle ⇒ Separable: Non‐separable = entangled: MOST states are entangled!! Two‐level systems 2‐level systems: atom, quantum circuits, spin, polarizaon of photon Rotaon (e.g. polarizer, Raman, microwaves…) Rabi frequency Bell states What is so special about entanglement? |↑〉 |↑〉 |↓〉 |↓〉 1000 km A B 1. Perfect correlaons (even at very large distance…!) Cond. proba. What is so special about entanglement? |↑〉 |↑〉 |↓〉 |↓〉 1000 km A B 1. Perfect correlaons (even at very large distance…!) 2. You CAN NOT assigne a state to A or B = system as a whole Stascal mixture 3. You CAN NOT “dis‐entangled” by a LOCAL rotaon Non Local Quantum correlaons Classificaon of entangled states Biparte Pure state Mixed state Mulparte Pure state Bi‐separable Fully entangled: not bi‐separable w/r to any bi‐paron of the system Examples of inequivalent entangled states GHZ‐state W‐state No LOCAL rotaon can transform GHZ state into W state ⇒ defines 2 inequivalent classes of entanglement 2 systems 1 classe 3 systems 2 classes PRA 62, 062314 (2000) 4 systems 9 classes PRA 65, 052112 (2002) Beyond …? Preparaon of entangled states (1) 1. Conservaon laws Atomic cascade (e.g. Ca) Radio‐acve decay J=0 J=1 J=0 2. Some “symmetry”: e.g. idencal parcles Bosons 1 and 2 in two states and Fermions: Pauli principle Preparaon of entangled states (2) 3. Interacon between the sub‐parts evoluon Examples of Hint: Spin orbit coupling Hyperfine structure (ground state of H) Magnec interacon Generic Heisenberg Hint: e.g. spin 1/2 J Generaon of “cluster” states: (Ising) Preparaon of entangled states (3) 4. Projecve measurement Prepare Apply “small” rotaon Non‐destrucve measurement of (e.g. with cavity) Project onto: Heralded entanglement (probability ε2) Bi‐parte entanglement and non‐locality Bell inequality tests (1964) x S y x Entangled state ⇒ Classical local correlaons ⇒ y Non‐locality t ⇒ A and B not causaly related B A x Entanglement between A and B possible Quanficaon of entanglement: quantum state tomography For N atoms where (4N‐1 coefficients) Measure ⇒ reconstruct the density matrix e.g. Bell states of 2 ions PRL 92, 220402 (2004) Le largest state characterized: 8 ions W states, 65531 coefficients 10 hours of measurements! Too hard for “large” systems Nature 438, 643 (2005) Quanficaon of entanglement (1) Biparte ~OK: Schmidt decomposion, entropy, concurrence… Mulparte Pure states: generalized Bell inequalies (PRL 104, 240502 (2010)) Mixed states: no general criteria (even if you know ρ) ⇒ use weaker criteria Fidelity: want , measure Separable state ⇒ C.A. Sacke et al., Nature 404, 256 (2000) Quantum non‐locality (Bell for 2 at.) 0 1/2 1 Separable entangled Quanficaon of entanglement (2) More generally: entanglement witness operator separable state ⇒ ⇒ entangled state Examples: 1. fidelity ⇒ 2. Energy of spin 1/2: Separable: ⇒ Singlet state: ⇒ Choose Useful for macroscopic measurements Entanglement is fragile… Atom loss N‐1 GHZ strongly correlated While: W weakly correlated Phase fluctuaons (fluctuang B, E field) and Decoherence faster when N large Decoherence of a GHZ state PRL 106, 130506 (2011) N‐ion string But you can protect it…! Start from Energy 0 Energy Apply local rotaon on “atom” B and D (does not change class of entanglement) evoluon insensive to ⇒ fluctuaon of ω0 Decoherence free subspace Entanglement, measurement and classicality System Environment S‐M Meter M‐E α|g〉 + β |e〉 Entanglement S – M – E Stascal mixture: get |g〉 (proba |α|2) OR |g〉 (proba |α|2) Entanglement = resource for quantum metrology Frequency standard |F=4〉 ω0 = 9 192 631 770 Hz 133Cs |F=3〉 Applicaons: test of relavity, navigaon using GPS… Today, best atomic clock (ion based) Δν/ν ~ 10‐17 We can do beer using entanglement! Prepare N atom in Pe more sensive measurement t Entanglement = resource for quantum computaon Help solving “hard” problems: factoring (Shor) Exp L searching (Grover) Ln Encode informaon on a qubit: |0 , |1 〉 〉 calculation Time size L Elementary bricks (“circuit” approach): One‐qubit gate U1 c Two‐qubit gate U t 2 C‐NOT Uf Highly N qubits interfere entangled & entangle Engineered entangled states: state‐of‐the art (2013) 14 ions, GHZ (F=0.51) PRL 106, 130506 (2011) 3 supracond. qubits W (F=0.78), GHZ (F=0.62) Nature 467, 570, 574 (2010) 3 Rydberg atom GHZ (F=0.57) Science 288, 2024 (2000) 2 ground‐state atoms ~ Bell (F~0.75) 50 atoms cluster state Mandel et al., 425, 937 (2003) PRL 104, 010502, 010503 (2010) 8 photons (H,V) GHZ (F=0.71) Nat. Phot. 6, 227(2012) A short bibliography on entanglement Theory (from “easy” to hard…) “Exploring the quantum”, S. Haroche and J‐M. Raimond Cambridge “Quantum compung and entanglement”, D. Bruss and C. Machiavello Les Houches summer school 2009 (Singapore) “Experimental procedures for entanglement verificaon”, PRA 75, 052318 (2007) “Entanglement in many body system”, RMP 80, 517 (2008) Experimental papers on quanficaon of entanglement Q.A. Turchee et al., PRL 81, 3631 (1998) C.A. Sacke et al., Nature 404, 256 (2000) D. Leibfried et al., Nature 438, 639 (2005) Review on Rydberg and QIP Saffman RMP 82, 2313 (2010) Outline 1. Basics of entanglement 2. Entanglement and 2‐atom gate using Rybderg blockade Trapping two atoms in tweezers: well‐defined geometry Dipole trap A lens Atom B 2 – 20 µm Atom A Vacuum See talk YEA H. Labuhn Dipole trap B One counter / atom A & B Atom A Fluorescence (780 nm) Atom B Two‐photon Rydberg excitaon (87Rb): well‐defined states 58d3/2 F = 3 σ+, 475 nm 297 nm Δ =720 MHz 5p1/2 F = 2 π, 795 nm See talk YEA H. Labuhn 5s1/2 Detect atom loss = Rydberg not trapped Miroshnychenko et al., PRA 82, 023623 (2010) vdW interacon between 2 atoms 2 atoms A and B with states ns, (n‐1)p and np; distance R np, (n‐1)p δ(n) ns, ns |ns,ns〉’ Ex.: n = 60 δ= 5‐10 GHz, ⇒ 3 C3/R ≈ 100 MHz In the vdW approximaon: shi of |r,r〉 only… A note on vdW interacon between 2 atoms: role of symetry There are 3 two‐atom states! np, (n‐1)p (n‐1)p, np δ(n) ns, ns A note on vdW interacon between 2 atoms: role of symetry There are 3 two‐atom states! δ(n) 0 ns, ns A note on vdW interacon between 2 atoms: role of symetry There are 3 two‐atom states! δ(n) ns, ns Sll two‐state model, but: 6 Typical interacon strength: C6(43s) ≈ ‐ 2.4 GHz.µm 6 C6(53d) ≈ 15 GHz.µm 6 C6(82d) ≈ 8500 GHz.µm ⇒ Vvdw ~ 1 ‐ 100 MHz for R ~ few µm Rydberg blockade: “addressable” version (U. Wisconsin) 2E |r ,r R A B〉 A B E |rA,gB〉 |gA,rB〉 2 atom energy 0 |gA,gB〉 Rydberg blockade: “addressable” version (U. Wisconsin) |rA,rB〉 2E R A B E |rA,gB〉 |gA,rB〉 2 atom energy 0 |gA,gB〉 Rydberg blockade: “addressable” version (U. Wisconsin) |rA,rB〉 2E R A B E |rA,gB〉 |gA,rB〉 2 atom energy Oscillaon of atom B 0 |gA,gB〉 E. Urban et al., Nat. Phys. 5, 110 (2009) 90d5/2 Rydberg blockade: “addressable” version (U. Wisconsin) |rA,rB〉 2E R A B E |rA,gB〉 |gA,rB〉 2 atom energy Oscillaon of atom B 0 |gA,gB〉 E. Urban et al., Nat. Phys. 5, 110 (2009) 90d5/2 Rydberg blockade: “addressable” version The “blockade radius” picture Rabi formula: |rB〉 VvdW Defines Rb: Pr,max |gB〉 Atome B when A in |rA〉 R Blockade alone does not provide entanglement : need indisnguishability w/r excitaon ⇒ collecve excitaon Rydberg blockade: collecve excitaon (IO Palaiseau) A |r ,r A B〉 R 2E B E |rA,gB〉 |gA,rB〉 2 atom energy 0 |gA,gB〉 Rydberg blockade: collecve excitaon (IO Palaiseau) A |r ,r A B〉 R 2E B E 2 atom energy 0 |gA,gB〉 Rydberg blockade: collecve excitaon (IO Palaiseau) A |r ,r A B〉 R 2E B E 2 atom energy 0 |gA,gB〉 Recall (RW approx.): Rydberg blockade: collecve excitaon (IO Palaiseau) A |r ,r A B〉 R 2E B 0 E 2 atom energy 0 0 |gA,gB〉 Recall (RW approx.): Rydberg blockade: collecve excitaon (IO Palaiseau) A Collecve 2‐level system R 2E B E 2 atom energy 0 1.0 |gA,gB〉 0.8 0.6 0.4 A. Gaëtan et al., Nat. Phys. 5, 115 (2009) Excitation probability 0.2 0.0 0 40 80 120 160 200 240 280 320 Duration of the excitation (ns) Rydberg blockade: collecve excitaon (IO Palaiseau) A Collecve 2‐level system R 2E B E 2 atom energy 0 1.0 |gA,gB〉 0.8 0.6 0.4 A. Gaëtan et al., Nat. Phys. 5, 115 (2009) Excitation probability 0.2 0.0 0 40 80 120 160 200 240 280 320 Duration of the excitation (ns) Rydberg blockade: collecve excitaon (IO Palaiseau) If rA and rB fluctuate from shot‐to‐shot: not entangled!! 2E E 2 atom energy 0 1.0 |gA,gB〉 0.8 0.6 0.4 A. Gaëtan et al., Nat. Phys. 5, 115 (2009) Excitation probability 0.2 0.0 0 40 80 120 160 200 240 280 320 Duration of the excitation (ns) Entangled of two atoms using the Rydberg blockade Collecve excitaon version rA rB 4 µm A B ⇒ with If atomic moon frozen ⇒ Analyzing entanglement Measure the density matrix Extract the fidelity Raman rotaon: check coherence of the superposion 100 Δ 5p1/2 80 60 795 nm 40 Population in F = 1 (%) 20 |↑〉 0 0 1 2 3 4 5 6 7 8 9 10 Duration of Raman pulse (µs) |↓〉 Global rotaon on the two atoms: 2 × ΩRaman ⇒ Rotaon ⇒ transformes coherence into populaon Measurement of entanglement 0.5 t) 0.4 Ram Ω 11 ( ↓↓ 0.3 0.2 Probability P 0.1 Probability P 0.0 0.0 0.5 1.0 1.5 2.0 ΩRamt / 2π Measurement of entanglement 0.5 t) 0.4 Ram Ω 11 ( ↓↓ 0.3 0.2 Probability P 0.1 Probability P 0.0 0.0 0.5 1.0 1.5 2.0 ΩRamt / 2π Details in Gaëtan et al., NJP 12, 065040 (2010) Atom losses during the entangling sequence Entangling sequence Losses 38% Pairs of atoms 62% Ftotal = 0.46 Fpairs = Ftot / 0.62 Etc… Etc… Pairs of atoms Losses Fpairs = 0.75 ± 0.07 Paral state reconstrucon and error budget State prepared: Aer loss correcon: Error budget Fpairs = 0.75: Imperfect blockade 0.12 Technical (laser stability, sp.