Entanglement

and

Rydberg interacon

Antoine Browaeys, Instut d’Opque, CNRS Outline

1. Basics of entanglement

2. Entanglement and 2‐ 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) (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

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 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 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‐ 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 : |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 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 (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 2 atom energy 2E 0 E Rydberg blockade: “addressable” version (U. Wisconsin) |r A ,g B 〉

|g A ,g B 〉

|r A ,r B 〉

|g A ,r B 〉

A R B 2 atom energy 2E 0 E Rydberg blockade: “addressable” version (U. Wisconsin) |r A ,g B 〉

|g A ,g B 〉

|r A ,r B 〉

|g A ,r B 〉

A R B 2 atom energy 2E E. Urban 0 E Rydberg blockade: “addressable” version (U. Wisconsin) et al. |r A ,g , Nat. Phys B 〉

. 5 , 110 (2009) |g A ,g B 〉

|r A ,r B 〉

90d |g 5/2 A Oscillaon of atom B ,r B 〉

A R B 2 atom energy 2E E. Urban 0 E Rydberg blockade: “addressable” version (U. Wisconsin) et al. |r A ,g , Nat. Phys B 〉

. 5 , 110 (2009) |g A ,g B 〉

|r A ,r B 〉

90d |g 5/2 A Oscillaon of atom B ,r B 〉

A R B 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〉 2 atom energy 2E 0 E Rydberg blockade: collecve excitaon (IO Palaiseau) |g A ,g B 〉

|r A ,r B 〉

A B R 2 atom energy 2E Recall (RW approx.): 0 E Rydberg blockade: collecve excitaon (IO Palaiseau) |g A ,g B 〉

|r A ,r B 〉

A B R 2 atom energy 2E Recall (RW approx.): 0 E Rydberg blockade: collecve excitaon (IO Palaiseau) |g A ,g B 〉

0 0 |r A ,r B 〉

A B R 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. emission) 0.1 Moon of the atoms 0.03 Wilk et al., PRL 104, 010502 (2010) Influence of the moon of the atoms: how much frozen?

1.0

↑↓

P 0.5 -

↓↑ P -

0.0 ↑↑ P + -0.5 = P00+P11-P01-P10 ↓↓ P Π

-1.0 0 1 2 3 4 5 6 Duree Raman pulse (µs) Influence of the moon of the atoms: how much frozen?

1.0

↑↓

P 0.5 -

↓↑ P -

0.0 ↑↑ P + -0.5 = P00+P11-P01-P10 ↓↓ P Π

-1.0 0 1 2 3 4 5 6 Duree Raman pulse (µs)

Coherence reduced by 0.5

Theory: Quantum gate using Rydberg blockade

D. Jacksch et al., PRL 85, 2208 (2000)

Sequence:

πA – 2πB – πA control target Table of truth: 11 → 11 → 11 → 11 10 → 1r → 1r → ‐10 01 → r0 → r0 → ‐01 00 → r0 → r0 → ‐00 Blockade

From π‐gate to CNOT Uπ 11 → 10 c 10 → 11 01 → 01 t H H 00 → 00 The CNOT gate at uni. Wisconsin (1)

Table of truth: check blockade

F = 0.73

Isenhower et al., PRL 104, 010503 (2010) The CNOT gate at uni. Wisconsin (2)

Prepare entangled states

Prepare (|0〉+|1〉)|0〉 gate

00 10 01 11 Check coherence: global Raman rotaon

Amplitude ⇒

F = 0.58 (with loss correcon) See also: Zhang et al., PRA 82, 030306(R) (2010) Conclusion on the gate and entanglement

Proof‐of‐principle: OK

But needs to improve fidelity!!

And careful with the phases…