Tutorial: Quantum Information & Cold Atoms

Tutorial: 

Quantum Information & Cold Atoms 

• cold atoms as a tool in quantum information 

• applications: 

• quantum computing 

• quantum communications 

• precision measurements 

Peter Zoller 

Institute for Theoretical Physics 

University of Innsbruck Austria 

1

Entangled States 

• entanglement 

A B 

states: 

but also ... 

|0 |0 

|1 |1 

1 

2 

|0 |0 |1 |1 

... product states 

... entangled 

• fundamental aspects of quantum mechanics 

- incompatibility of QM with LHVT 

- decoherence 

- measurement theory (?) 

• applications 

- quantum communications & computing 

- precision measurement 

Schrödinger: 

Verschränkung 

2

Engineering Entangled States 

We need ... 

• “quantum engineering” 

A B environment 

A B 

|aA|bB cab|aA|bB 

Hamiltonian evolution 

• or: “quantum gambling” 

measurement 

• isolation 

|A|B|E |ABE 

AB trE|ABE| 

|AB| 

Quantum optical systems provide one of the best set-ups to create 

entangled states in a controlled way. 

3

Quantum Optics Road Map 

• Starting from the quantum optical tool box: 

photons 

... manipulate & couple on level of single quanta 

atoms 

• we assemble large systems ... 

node 

channel 

• Nodes: local quantum computing 

- store quantum information 

- local quantum processing 

• Channels: quantum communication 

- transmit quantum information 

4

Quantum information processing 

• quantum computing 

ouput 

quantum 

processor 

input 

• quantum communications 

| 

| | 

| 

|out 

|out Û|in 

|in 

| 

transmission of a quantum state 

quantum weirdness: 

superposition 

entanglement 

interference 

nonclonability and 

uncertainty 

no decoherence! 

teleportation 

crytography 

5

Qubits, Quantum Gates etc. 

• quantum bits or qubits 

n spin1/2 

• single qubit gate 

• two qubit gate 

|1i 

|0i 

example: two qubits 

| Û1| U1 

control target 

| c00|00 c01|01 c10|10 c11|11 

|a 

|b 

|a 

Controlled-NOT 

|a b 

6

1. Cold atoms as quantum memory 

• technique; cooling and trapping of atoms 

• decoherence properties 

2. How to entangle atoms? 

7

1. Cold atoms as quantum memory 

• cold atoms • single qubit gates 

single trapped atom: 

|0 |1 

qubit in longlived 

internal states 

trap 

addressing 

single qubit 

laser 

laser 

|0 |1 

8

Remarks: Traps ... 

• ion traps 

200 µm 

Boulder linear ion trap 

NIST Boulder, Innsbruck, Munich, 

Hamburg, Aarhus, Oxford, London, ... 

ions 

issues: 

conservative potential 

(heating?) 

single atom loading 

laser cooling to ground 

state 

9


• far-offresonance optical lattice 

nonresonant 

laser 

arrays of microtraps 

AC Stark shift 

laser 

optical lattice 

issues: 

linear ion trap 


single atom loading (?!) 

laser cooling 

laser 

10


• loading a lattice from a Bose Einstein condensate 

– regular filling with exactly 1, 2 or 3 atoms per lattice site via Mott insulator 

quantum phase transition (Bose Hubbard model) 


– large number of atoms 

Bose Hubbard model 

• superfluid phase 

• Mott insulator 

theory: Innsbruck 1998 

exp: Munich 2001 

. . . . 

. . . . 

 

b1 b2 bM |vac 

"Fock states" 

11


• single atom FORTs 

array of FORTs (Hannover) 

two movable single-atom 

FORTs (Orsay) 

4 µm 

12


• magnetic traps 

atom "chips" 

Schmiedmayer 

Heidelberg, Munich, 

Harvard, Orsay 

issues: 



surface effects (?) 

single atom loading (?) 

laser cooling (?) 

13

... and tricks with traps 

• We can move atoms around 

|qubit |motion 

• optical traps: internal state 

dependent potentials 

|0 |1 

qubit 

fine structure 

Alkali atom 

|0 

|1 

14

2. Engineering entanglement: two-qubit gates 

• implement entanglement of two qubits 

U2 

• How? 

example: 

phase gate 

auxiliary collective mode as data bus 

controllable two body interactions 

dynamical phases 

geometric phases 

|00 |00 

|01 |01 

|10 |10 

|11 e i |11 

difficult 

15

Concept 1: two-qubit gates via quantum databus 

- Entanglement via collective auxiliary quantum degree of freedom 

Collective mode 

1 2 

qubits 

quantum 

data bus 

switch 

Examples: 

Ion traps 

Cavity QED 

state vector: 

| cx|xN1xN2 x0 |collective mode 

x 

gate: 

quantum register data bus 

swap 

requirement: cooling of the collective mode (= prepare a pure state) 

16

Ion Trap Quantum Computer 

• Cold ions in a linear trap 

laser pulses entangle ion pairs 

• State vector 

Blatt 

|Ψi = X cx|xN−1,... ,x0iat om |0iphonon 

quantum register databus 

theory: Innsbruck, Aarhus, London, Brisbane ... 

exp: NIST Boulder, Innsbruck, Munich, Oxford .. 

• Qubits: internal atomic states 

• Quantum gates: entanglement 

via exchange of phonons of 

quantized center-of-mass mode 

• Achievements: 

– entanglement of four ions 

– Bell measurements 

– individual addressing 

– ground state laser cooling 

17

Optical Cavity QED 

• optical photons in a high-Q cavity as "data bus" 

laser 

spontaneous 

emission 

cavity decay 

theory: Innsbruck, London ... 

exp: Caltech, Georgia Tech, Munich 

laser 

cavity 

adiabatic pasage 

decoherence free subspace 

18

Concept 2: two-qubit gates via two-body interactions 

- Controlled two-body interaction 

V(R) 

1 2 

qubits 

• physical mechanisms 

We must design a Hamiltonian 

so that 

optical dipole – dipole (Albuquerque) 

cold collisions (Innsbruck) 

H Et|1 11| |121| 

|11 |12 e i |11 |12 

Rydberg – Rydberg (Innsbruck + Harvard + Storrs) 

19

Cold controlled collisions 

• consider two atoms in different internal states stored e.g. in an optical 

lattice 

• we move the lattice in a state dependent way, and induce a collision: 

the collision is sufficiently slow not to excite oscillations 

before ... 

during ... 

after collision 

atom 1 atom 2 

• phase gate via collisions ... 

cold atoms 

on site interaction = 

phase shift 

(coherent collision) 

20

Entanglement 

• parallelism 

e 

e 

• fidelity F~99% 

• ... slow 

i φ 

1 0 1 0 

i φ 

i φ 

e 

e i φ i φ 

e 

1 0 1 0 1 0 1 0 1 0 

Lift 

Shift 

Lower 

21

Rydberg – Rydberg interactions 

• Rydberg atom in constant electric field 

• setup 

atom 

E 

energy 

• linear Stark effect 

n ~ 15 

• Large dipole-dipole interaction 

µ 1 

Vdipr 

µ 2 

|r 

laser 

|g 

E 

theory: Harvard + Innsbruck + 

Storrs, Aarhus 

• permanent dipole moment 

E 1kV/cm 

n 2 huge! 

R opt/2 300 nm 

E 60 GHz 

Vdip 4 GHz 

for n 15 

22

• 2 qubit quantum gate 

|00 |00 

|01 |01 

|10 |10 

|11 e i |11 

fast gate 

energy shift for time 

interval = phase 

23

"Dipole blockade" 

• atomic configuration 

Atom 1 Atom 2 

|r1 

1 

|11 

qubit 

|01 

Vdip u 

laser laser 

|1 2 

qubit 

|0 2 

2 

|r 2 

• dipole blockade 

2 

| r0〉 

| 0r〉 

1 

| rr〉 

| 00〉 

Vdip u 

Large dipole-dipole 

interaction shifts level 

off resonance: 

No double excitation - 

no force! 

24

3. Mesoscopic atomic ensembles 

• idea 

– dipole blockade mechanism 

theory: Harvard + Innsbruck, exp: Storrs, ... 

25

Configuration 

• mesoscopic atomic ensembles (instead of microscopic quantum 

objects) 

- coherent manipulation of collective excitations of atomic ensembles 

10 m 

-underlying physics: 

dipole blockade 

N 100 atoms 

laser 

ground state 

|g 

|r 

laser 

|q 

26

Manipulating collective excitations 

• ground state 

|g N |g1|g2 |gN 

• one excitation (Fock state) 

|g N1 q i |g 1 |qi |g N 

• two excitations 

|g N2 q i,j |g 1 |qi |q j |gN 

laser 

etc. 

We can store and manipulate qubits. 

laser 

|g 

|r 

|q 

blockade 

27

• actual configuration 

hopping via resonant dipole-dipole interaction 

atom i atom j 

|g 

|p 

|r 

|p 

|p 

|r 

|p 

|q |q 

|g 

resonance condition: Er E p E p Er 

idea: Cote, Lukin 

28

cont. 

• qubits 

| |g N |g N1 q 

• entanglement of ensembles 

+ 

superposition 

29

4. Quantum Communication 

• Question: 

– what can we do with 10 qubit quantum computers? 

• Example: 

– quantum communication with memory and quantum error 

correction = quantum repeater 

30

Quantum Communications 

• classical communications • quantum communications 

0 0 

Alice Bob Alice Bob 

0 0 

1 

1 

0 

0 

1 

|Ψ1i|Ψ2i 

networks 

cryptopgraphy 

|Ψ3i |Ψ4i 

|Ψ5i|Ψ6i 

Alice Bob Alice Bob 

1 0 

0 

0 

1 

|Ψ1i|Ψ2i 

ρ ρ 

Eve Eve 

ρ ρ 

|Ψ4i 

31

Optical Interconnects 

• A cavity QED implementation 

Optical cavities connected by a quantum channel 

Node A Node B 

Laser 

• memory: 

atoms 

fiber 

• databus: 

photons 

• memory: 

atoms 

Laser 

32

Quantum Communication Protocols 

• Problem: the basic problem of quantum communication is noise / 

decoherence 

Alice Bob 

|φi 

A 

noise 

ρ 6= |φihφ| 

pure state mixed state: 

fidelity F || 1 

• Solution: quantum communication in the presence of noise is based on 

- teleportation 

- purification (error correction) 

33

1. Teleportation 

Bennett et al. PRL '93; exp with photons: Innsbruck, Rome, Caltech 

|φi 

C 

Alice Bob 

A 

noise 

|AB |0A|1B |1A|0B (singlet) 

EPR pair 

• Assume: Alice and Bob share a singlet state (given resource) 

• Idea: Alice can transmit a qubit to Bob without physically sending the 

qubit 

B 

34

Teleportation protocol: 

1. Alice performs a CNOT followed by a measurement of A and C 

|φi 

C A 

CNOT + state measurement 

2. Alice tells Bob the measurement outcome 

3. Rotation 

C 

C 

A 

A 

classical communication 

The state has been teleported to B. 

How do we get the EPR pair? 

⎧ 

⎪⎨ 

4 possible outcomes = 

⎪⎩ 

∀|φi with prob 1 

4 

B 

|φi 

B 

00 

01 

10 

11 

unitary operation 

35

2. Purifying EPR pairs 

• A noisy quantum channel allows us to generate many noisy EPR pairs. 

1/2 < F < 1 

... 

Bennett et al., Deutsch et al. PRL '95 

purification 

F ~ 1 

If F 1/2 we can purify and obtain one EPR pair with F 1. 

To do this we need a small quantum computer. 

• In summary: quantum communication via a noisy channel: 

- we generate one high fidelity EPR pair by purification 

- we teleport the quantum state 

36

3. Quantum Repeater 

• classical repeater 

"1" 

"0" 

A 

classical 

signal 

L0 

C 1 

absorption 

& dispersion 

A B 

• quantum repeater: 

– we cannot clone a quantum state!? 

– Idea: purify segments of EPR pairs ... and teleport 

L0 

C 2 

H. Briegel et al. PRL '98 

37

cont. 

• Quantum repeater protocol: generate a long distance EPR pair 

1/2 < F

Other (easier?) implementations 

• single atoms, single photons and high-Q cavities: "quantum engineering" 

Node A Node B 

Laser 

• memory: 

atoms 

fiber 

• databus: 

photons 

• memory: 

atoms 

Laser 

• atomic ensembles, free space, or low-Q cavities: "quantum gambling" 

Sample A 

Sample B 

deterministic 

strong coupling on 

single quantum level 

difficult ! 

probabilistic 

scheme 

noise tolerant 

linear optics 

simpler ! 

39

To illustrate the ideas ... 

Single Atoms 

laser 

laser 

ion A 

ion B 

Internal states 

ionA ionB 

- Weak (short) laser pulse, so that the excitation probability is small. 

- If no detection, pump back and start again. 

|1i |1i 

|0i |0i 

- If detection, an entangled state is created.because we do not know 

which atom emitted the photon 

|0, 1i±|1, 0i 

40

Atomic Ensembles 

• system: cloud of cold atoms 

laser Stokes 

ensemble 

• Raman process: 

|0 N |vac (atomic ground state) 

1 

Na 

Na 

 

i1 

• state of atomic collective mode + Stokes photon 

Internal states 

laser 

|0i 

|r 

Stokes 

|1i 

|01 1i 0Na a |vac (single atomic excitation) 

| |vac pc a 

cStokes|vac O pc 

a,a 1 (for weak excitation) 

2 pc 1 

... analogous to parametric 

downconversion 

41

Generation of Entanglement (Ensembles) 

Sample A 

Sample B 

|A |B |vacA pc a c s |vacA |vacB pc b c s |vacB 

measurement gives 

|AB a b |vac 

|1a,0b |0a,1b 

We have generated entanglement between collective atomic states 

42

Continuous Variable Teleportation 

• Instead of qubits we consider now continuous variable quantum states 

| dx|x x x position 

p momentum 

x,p i 

• transmission of a cv state 

|φi |φi |φi 

• continuous variable teleportation 

|φi 

C A 

EPR pair 

|φi 

Alice Bob 

|EPRAB dx |xA|xB 

dp |pA| pB 

|φi 

B 

Vaidman 

Braunstein 

Kimble (exp) 

x A x B|EPR x 1|EPR 

p A p B|EPR p 1|EPR 

43

Teleportation with Squeezed Light 

• Two-mode squeezed light: 

electric field 

• Scheme 

S. Braunstein, H.J. Kimble et al., PRL '98; Science '99 

E ae ikxit 

x ip 

quadrature components 

Bell 

3 

|φi 

out 

in 1 2 

pump squeezed light 

=EPR state 

2 

parametric 

downconversion 

EPR source 

squeezed light 

1 

2 

44

Atomic ensembles as quantum memory for cont var states 

• We consider an ensemble of N atoms 

ensemble of 

atoms 

two-level atom 

=spin -1/2 

• a collection of two-level atoms can be described in terms of a collective 

“angular momentum“ 

~S a = 

collective angular 

momentum 

NX 

µ=1 

1 

2 ~σ(µ) 

two-level atom 

= spin -1/2 

1 2 

Bloch vector 

x 

z 

45 

y

atoms cont. 

• superposition of the two ground states: coherent spin state 

1 2 

Bloch vector 

• quantum fluctuations 

S y a , Sz a iSx a 

S y a Sz a 1 

2 |S x a | 

we treat S x a classically and rescale 

X a , P a i X a P a 1 

2 

canonical communation relations 

1 

2 

|1 |2 N 

S a Sx a , Sy a , Sz a N a 

2 ,0,0 

x 

P a 

Bloch picture 

z 

X a 

| 46 

a dXa |XaX a y 

coherent spin state =vacuum state 

there are many cv quantum states 

around it:

Teleportation with coherent light + atomic ensembles 

coherent 

light 

ex 

atoms 1 atoms 2 

in 

out 

measurement projects atomic ensembles 

into continuous variable EPR state 

|EPR dP|P A| PB dX |XA|XB 

theory: Innsbruck 

EPR 

experiment: E. Polzik et al. (Aarhus), Nature 2001 

measure 

47

Atomic ensembles: quantum memory for light 

• purpose 

incoming light pulse 

unknown (arbitrary) state 

known shape of wave 

packet 

• how? example ... 

theory Harvard, Aarhus 

exp: Harvard 

write 

[ Atomic ensemble ] 

storage medium outgoing light pulse 

cavity 

a1 

atoms 

read 

cavity laser 

same state 

reshaping 

48

4. Precision measurements 

• maximally entangled states and squeezed atomic states for precision 

measurement beyond the standard quantum limit 

theory: NIST Boulder, Innsbruck, Aarhus, Georgia Tech, Harvard 

exp: NIST Boulder, Aarhus, Rochester 

49

Ramsey method 

|0 |1 

π / 

π / 

2 

2 

#ofatomsin|0 

|0 1 

2 

P0T cos 1 

2 L 0T 2 

|0 |1 

1 

2 |0 e i L 0t |1 

cos 1 

2 L 0t|0 sin 1 

2 L 0t 

y 

 

/2 

Nb 

/2 

0T 

x 

L 0T 

x 

x 

Bloch picture 

z 

|0 

z 

y 

y 

y 

50

We compare ... 

• N independent atoms 

SQL 

1 

T nrep 

1 

N 

standard quantum noise limit 

Remarks: 

N is limited: density / collisions 

T is limited: decoherence 

• N entangled atoms 

ent 

• figure of merit: to achieve the same uncertainty ... 

2 nrep SQL 

nrep ent 

N 

fN 2 1 N 

1 

T nrep 

• present experiments: NIST Boulder 4 ions entangled 

1 

fN 

1 

T nrep 

|0000 |1111 

1 

N 

Heisenberg limit: 

maximally entangled state 

We want 2 1 

51

Jx 

Example: Spin Squeezing with BEC 

• Entanglement via collisions 

BEC 

|0 N 

product 

state 

Jz 

N 

laser 

• Bloch vector picture 

Jy 

# atoms N 

Jx 

[ 1 

2 

(|0 |1 N 

 

product 

state 

Jz 

collisions 

Jx 

Jy 

theory: Aarhus + Innsbruck, Georgia Tech, 

Harvard; exp.: see also Yale 

cn|0 n |1 Nn 

entangled state 

Jz 

squeezing below the 

standard quantum limit 

Jy 

52

5. Topological quantum computing 

concept: Kitaev, Preskill, Freedman 

• engineering model systems with topologically protected ground states: 

indistinguishable under local operations 

• Fractional quantum Hall effect: electron in 2D + magnetic field: isolated 

ground state with gapped excitations 

• on a closed surface of genus g: renders the ground state degenerate. 

• transformation of degenerate ground states into each other via motion 

of flux quantum around nontrivial topological paths 

• Rem.: so far no atomic physics models 

53

A (small) first step: ½ - (abelian) anyons in small BECs 

• quantum degenerate Bose gas: 2D and rotating trap 

BEC 

Ω = 0 

1 vortex 2 vortices 

• what happens is a fractional quantum Hall scenario with bosons 

Ω 

rotate the trap 

? 

Ω 

Ω ∼ ω 

product states entangled states ☺ 

Laughlin fluid excitations: 

½ anyons 

laser 

we can create and 

manipulate anyons! 

54

1. System: N bosonic atoms 

• N bosons 

• Hamiltonian in rotating frame 

H = 1 

2 

NX 

µ 

−∇ 2 i + r2 i 

i=1 

Ω 

− 2 

ω Liz 

 

+ η 

NX 

δ(ri − rj) 

length scale: motion in trap rotation contact interaction 

` =(¯h/mω) 1/2 

• In the limit Ω ∼ ω equivalent to (fermionic) quantum Hall Hamiltonian 

Gunn & Wilkin '00 

Ω 

harmonic trapping potential 

rotation 

Ω 

i

2. Energy spectrum 

• independent atoms 

harmonic trap 

... 

... 

¯hω 

2¯hω 

Landau wave functions: 

z m e − 1 2 |z|2 

+ rotation 

x + iy 

• we assume: dynamics restricted to first Landau levels 

Ω 

... 

... 

¯h(ω − Ω) 

"Landau levels": 

~ degenerate 

kT ¿ ¯hω 

56

• interacting atoms 

– dynamics in first "Landau level" 

Vint 

collisional interaction 

... 

¯h(ω − Ω) 

rotation 

• for Ω ∼ ω a parameter regime of effectively strong interactions 

Vint À ¯h(Ω − ω) 

external parameter 

strongly correlated qu fluid 

dilute gas! 

57

cont. 

• ground state: ½ - Laughlin state 

physics: atoms want to be in a collisional dark state to minimize energy: 

built from lowest 

Landau levels 

ψ[z] = Y 

i

3. Excitations: ½ anyons 

• quasihole at position 

ψ z0 

z0 

[z] =Y(zi 

− z0) ψ[z] 

i 

half atom missing: 

½ anyons z0 

• creating a quasiparticle: piercing with am offresonant laser 

z0 

laser 

H◦ = HL + V0 

X 

δ(zi − z0) 

i 

off-resonant laser interaction 

59

4. Anyons & fractional statistics 

We adiabatically drag the quasihole around ... and pick up a Berry's phase 

• one anyon 

• two anyons 

ψ z0 −→ eiφ ψ z0 

Berry's phase 

φ =2πn # atoms inside the curve 

ψ z0z1 −→ eiφ ψ z0z1 

φ =2π(n − 

2 

1 )=2πn− π 

½ anyons: π/2 bosons: 

(fractional statistics) 

fermions 

π 

there is an extra phase 

independent of area 

if we interchange the two 

holes: π/2 

2π 

π 

Fractional statistics has never been seen directly! 

60

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