Entanglement
and
Rydberg interac on
Antoine Browaeys, Ins tut d’Op que, 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 interac on 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
Superposi on principle ⇒
Separable:
Non‐separable = entangled:
MOST states are entangled!! Two‐level systems
2‐level systems: atom, quantum circuits, spin, polariza on of photon
Rota on (e.g. polarizer, Raman, microwaves…)
Rabi frequency
Bell states What is so special about entanglement?
|↑〉 |↑〉
|↓〉 |↓〉 1000 km A B 1. Perfect correla ons (even at very large distance…!)
Cond. proba. What is so special about entanglement?
|↑〉 |↑〉
|↓〉 |↓〉 1000 km A B 1. Perfect correla ons (even at very large distance…!)
2. You CAN NOT assigne a state to A or B = system as a whole
Sta s cal mixture
3. You CAN NOT “dis‐entangled” by a LOCAL rota on Non Local Quantum correla ons Classifica on of entangled states
Bipar te
Pure state
Mixed state
Mul par te
Pure state
Bi‐separable
Fully entangled: not bi‐separable w/r to any bi‐par on of the system Examples of inequivalent entangled states
GHZ‐state
W‐state
No LOCAL rota on 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 …? Prepara on of entangled states (1)
1. Conserva on laws Atomic cascade (e.g. Ca) Radio‐ac ve decay J=0
J=1 J=0
2. Some “symmetry”: e.g. iden cal par cles Bosons 1 and 2 in two states and
Fermions: Pauli principle Prepara on of entangled states (2) 3. Interac on between the sub‐parts
evolu on
Examples of Hint: Spin orbit coupling Hyperfine structure (ground state of H)
Magne c interac on
Generic Heisenberg Hint: e.g. spin 1/2 J
Genera on of “cluster” states: (Ising) Prepara on of entangled states (3) 4. Projec ve measurement
Prepare
Apply “small” rota on
Non‐destruc ve measurement of (e.g. with cavity)
Project onto:
Heralded entanglement (probability ε2) Bi‐par te entanglement and non‐locality
Bell inequality tests (1964) x S y
x Entangled state ⇒ Classical local correla ons ⇒ y Non‐locality t ⇒ A and B not causaly related B
A x Entanglement between A and B possible Quan fica on 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) Quan fica on of entanglement (1)
Bipar te ~OK: Schmidt decomposi on, entropy, concurrence…
Mul par te Pure states: generalized Bell inequali es (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 Quan fica on 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 fluctua ons (fluctua ng 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 rota on on “atom” B and D (does not change class of entanglement)
evolu on insensi ve to ⇒ fluctua on of ω0 Decoherence free subspace Entanglement, measurement and classicality
System Environment
S‐M Meter M‐E α|g〉 + β |e〉
Entanglement S – M – E
Sta s cal 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〉 Applica ons: test of rela vity, naviga on using GPS… Today, best atomic clock (ion based) Δν/ν ~ 10‐17
We can do be er using entanglement!
Prepare N atom in
Pe more sensi ve measurement t Entanglement = resource for quantum computa on Help solving “hard” problems: factoring (Shor) Exp L searching (Grover) Ln Encode informa on 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 compu ng and entanglement”, D. Bruss and C. Machiavello Les Houches summer school 2009 (Singapore)
“Experimental procedures for entanglement verifica on”, PRA 75, 052318 (2007)
“Entanglement in many body system”, RMP 80, 517 (2008) Experimental papers on quan fica on of entanglement Q.A. Turche e 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 excita on (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 interac on 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 approxima on: shi of |r,r〉 only… A note on vdW interac on 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 interac on between 2 atoms: role of symetry There are 3 two‐atom states!
δ(n) 0 ns, ns A note on vdW interac on between 2 atoms: role of symetry There are 3 two‐atom states!
δ(n) ns, ns
S ll two‐state model, but:
6 Typical interac on 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 Oscilla on 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 Oscilla on 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 indis nguishability w/r excita on ⇒ collec ve excita on Rydberg blockade: collec ve excita on (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: collec ve excita on (IO Palaiseau) |g A ,g B 〉
|r A ,r B 〉
A B R 2 atom energy 2E Recall (RW approx.): 0 E Rydberg blockade: collec ve excita on (IO Palaiseau) |g A ,g B 〉
|r A ,r B 〉
A B R 2 atom energy 2E Recall (RW approx.): 0 E Rydberg blockade: collec ve excita on (IO Palaiseau) |g A ,g B 〉
0 0 |r A ,r B 〉
A B R Rydberg blockade: collec ve excita on (IO Palaiseau)
A Collec ve 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: collec ve excita on (IO Palaiseau)
A Collec ve 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: collec ve excita on (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 Collec ve excita on version
rA rB
4 µm
A B
⇒ with
If atomic mo on frozen ⇒ Analyzing entanglement
Measure the density matrix
Extract the fidelity Raman rota on: check coherence of the superposi on
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 rota on on the two atoms:
2 × ΩRaman ⇒
Rota on ⇒ transformes coherence into popula on 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 Par al state reconstruc on and error budget
State prepared:
A er loss correc on:
Error budget Fpairs = 0.75: Imperfect blockade 0.12 Technical (laser stability, sp. emission) 0.1 Mo on of the atoms 0.03 Wilk et al., PRL 104, 010502 (2010) Influence of the mo on 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 mo on 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 rota on
Amplitude ⇒
F = 0.58 (with loss correc on) 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…