Atomic entangled states with BECAtomic entangled states with BEC

SFB Coherent Control€U TMR

A. Sorensen

L. M. Duan

P. Zoller

J.I.C.

(Nature, February 2001)

KIAS, November 2001.

Entangled states of atomsEntangled states of atoms

Motivation:• Fundamental.

• Applications: - Secret communication - Computation - Atomic clocks

• NIST: 4 ions entangled.

• ENS: 3 neutral atoms entangled.

Experiments:

j ª i6= j '1 i ­ j '2 i ­ : : : j 'N i

'

E ' 4

E ' 3

E 103This talk: Bose-Einstein condensate.

OutlineOutline

1. Atomic clocks

2. Ramsey method

3. Spin squeezing

4. Spin squeezing with a BEC

5. Squeezing and atomic beams

6. Conclusions

1. Atomic clocks1. Atomic clocks

To measure time one needs a stable laser

click

The laser frequency must be the same for all clocks

click

click

Innsbruck

Seoul

The laser frequency must be constant in time

click

Solution: use atoms to lock a laser

detector

feed back

frequencyfixeduniversal

In practice:Neutral atoms ions

! L = ! 0 + ±!

! 0

! L

Independent atoms:

Entangled atoms:

• N is limited by the density (collisions).

• t is limited by the experiment/decoherence.

• We would like to decrease the number of repetitions (total time of the experiment).

Figure of merit:

• To achieve the same uncertainity:

We want

±! =1

tpn r e p

pN

±! e n t =1

tp n r e p f (N )

±! e n t = ±!

»2 ¿ 1

»2 =(n r e p )e n t

n r e p=

T e n t

T=

pN

f (N )

2. Ramsey method2. Ramsey method

# of atoms in |1>

single atom

single atom

single atom

j0i !1p2(j0i + j1i )

!1p2(j0i + e¡ i (! 0¡ ! L )t j1i )

P1 = cos2·12(! 0 ¡ ! L )t

¸

sin·12(! 0 ¡ ! L )t

¸j0i+ co s

·12(! 0 ¡ ! L ) t

¸j1i

• Fast pulse:

• Wait for a time T:

• Fast pulse:

• Measurement:

Independent atomsIndependent atoms

Number of atoms in state |1> according to the binomial distribution:

where

If we obtain n, we can then estimate

The error will be

If we repeat the procedure we will have:

Another way of looking at itAnother way of looking at it

J x

J y

J z

J x

J y

J z

Initial state: all atoms in |0> First Ramsey pulse:

J x

J y

J z

J x

J y

J z

Free evolution:

J x

J y

J z

Measurement:

In generalIn general

where the J‘s are angular momentum operators

Remarks:

• We want

• Optimal:

• If then the atoms are entangled.

That is,

measures the entanglement between the atoms

»2 =N (¢ J z )2

hJ x i 2 + hJ y i 2

J ® =NX

k=1

j (k)®

»2 ¿ 1

»2 ¸ 1=N

»2 < 1

½ 6=X

n

pn½1 ­ ½2 ­ : : :½N

»2

J x

J y

J z

3. Spin squeezing3. Spin squeezing

No gain!

·1p2(j0i + j1i )

­̧ N

hJ x i = N=2 ¢ J x = 0

¢ J y = ¢ J z =pN =2

»2 =N (¢ J z )2

hJ x i 2 + hJ y i 2= 1

J x

J y

J z

• Product states:

hJ y i = hJ z i = 0

• Spin squeezed states:(Wineland et al,1991)

These states give better precission in atomic clocks

hJ x i ' N =2

¢ J z <pN =2

hJ y i = hJ z i = 0

»2 =N (¢ J z )2

hJ x i 2 + hJ y i 2< 1

How to generate spin squeezed states?How to generate spin squeezed states?

(Kitagawa and Ueda, 1993)

1) Hamiltonian:

It is like a torsion

Ât '1

2N 2=3

t=0 »2=1

»2 '1

N 2=3

H = ÂJ 2z

U = e¡ i (ÂtJ z )J z

»2

»2m in » N ¡ 2=3

ÂtÂtmin» 1=2N 2=3

1

2) Hamiltonian:

t=0 »2=1

Ât'1N

»2 '1N

Ât ' 1 jª i '1p2(j0; : : : ; 0i + j1; : : : ; 1i )

H = Â(J 2z ¡ J 2

y )

»2

Ât

1

»2m i n » N ¡ 1

Âtmin» 1=2N

ExplanationExplanation

Hamiltonian 1:

Hamiltonian 2:

J x ' N =2"

J ypN=2

;J zpN=2

#

= iJ xN =2

' i

X ´J ypN=2

H = ÂJ 2z =

ÂN2

P 2

H = Â(J 2z ¡ J 2

y ) =ÂN2

¡P 2 ¡ X 2

¢

P ´J zpN=2

t = 0

t = 0

t > 0

t > 0

ª (x; 0) / e¡ x2

are like position and momentum operators

4. Spin squeezinig with a BEC4. Spin squeezinig with a BEC

• Weakly interacting two component BEC

• Atomic configuration• optical trap

A. Sorensen, L.M Duan, J.I. Cirac and P. Zoller, Nature 409, 63 (2001)

laser

trap

F 1| 1

|0| 1

aaa! abb aab

AC Stark shift via laser:no collisions

H j a,b

d3r jr 2

2m 2 VTr jr

12 j a,b

U jj d3r jr jr jr jr

Uab d3r a r b r ar br

+ laser interactions

FORT as focused laser beam

Lit: JILA, ENS, MIT ...

a

b

A toy model: two modesA toy model: two modes

• we freeze the spatial wave function

• Hamiltonian

• Angular momentum representation • Schwinger representation

ax b x

ax axa bx bxb

spatial mode function

H 12Uaaa2a2 Uaba abb 1

2Ubbb2b2

ab ab

Jx 12 a b ab

Jy i2

a b ab

Jz 12

a a bb

H 12

Uaa Ubb 2UabJz2 Jx

= ÂJ 2z ¡ ­ J x

A more quantitative model ... including the motionA more quantitative model ... including the motion

• Beyond mean field: (Castin and Sinatra '00)wave function for a two-component condensate

with

• Variational equations of motion• the variances now involve integrals over the spatial wave functions: decoherence• Particle loss

| Na 0 Nb NNa

N

cNaNb |Na: aNa:t;Nb : bNb:t

d3x a x aNa : x, t

Na

Na !

d3x b x bNb : x, t

Nb

Nb!|vac

a

b

Time evolution of spin squeezingTime evolution of spin squeezing

• Idealized vs. realistic model • Effects of particle loss

1

10-1

10-2

10-3

10-4

0

4 8 12 16 20

t

2

idealized model

including motion

1

10-1

10-2

10-3

10-40 4 8 12 16 20

2

t 10X-4

loss

20 % loss

ideal

Can one reach the Heisenberg limit?Can one reach the Heisenberg limit?

H = ÂJ 2z ¡ ­ J x

H 2 = Â(J 2x ¡ J 2

z ) = Â(2J 2x + J 2

y ¡ J 2)

J 2x + J 2

y + J 2z = J 2 = constant

e¡ i ¼2 J xe¡ i±tJ 2z ei

¼2 J x| {z }e

¡ iÂ2±tJ 2x ' 1 ¡ i±t(2J 2

x + J 2y ) ' e¡ i±t(J 2

x ¡ J 2z )

e¡ i ±tJ 2y

­ t =¼2

t ¿ ±t

We have the Hamiltonian:

We would like to have:

| {z }

short pulseshort pulseshort evolution short evolution

Conditions:

Idea: Use short laser pulses.

H = ÂJ 2z

Stopping the evolutionStopping the evolution

»2

Ât

1

»2m i n » N ¡ 1

Âtmin» 1=2N

Once this point is reached, we wouldlike to supress the interaction

H = ÂJ 2z

The Hamiltonian is:

Using short laser pulses, we have an effective Hamiltonian:

J 2x + J 2

y + J 2z = J 2 = constant

In practice:In practice:

wait

short pulses

short pulse

5. Squeezing and entangled beams5. Squeezing and entangled beams

• Atom laser

• Squeezed atomic beam

• Limiting cases squeezing sequential pairs

• atomic configuration

collisional Hamiltonian

L.M Duan, A. Sorensen, I. Cirac and PZ, PRL '00

atoms

condensate as classical driving field

collisions

F 1| 1

|0| 1

condensate

Stark shift by laser:switch collisions onand off

pairs of atoms

1 x 1

x 02xe i2 t

1 x 1

x 02x

Equations ...Equations ...

• Hamiltonian: 1D model

• Heisenberg equations of motion: linear

• Remark: analogous to Bogoliubov

• Initial condition: all atoms in condensate

H i 1

i

x xx22m

Vx ixdx

gx, t 1

x 1 xe i2 t h.c. dx,

ix, t, jx , t ij x x

i t 1x, t xx22m

Vx 1x, t gx, t 1 x, te i2 t

i t 1 x, t xx2

2m Vx 1

x, t gx, t 1x, te i2 t

Case 1: squeezed beamsCase 1: squeezed beams

• Configuration

• Bogoliubov transformation

• Squeezing parameter r

• Exact solution in the steady state limit

B 1 1 Â 1 1 Â 1

B 1 1 Â 1

1 Â 1

tanhr | 1 || 1 |

| 1 |

| 1 |

g (x ,t)

0 a x

condensate

 1  1 B 1 B 1

input: vaccum

output

S q u eez ing p a ram e ter r v e rsu s d im ens io n le ss d e tu n in g /g 0 an d

in te rac tio n co effic ien t g 0 t

b ro ad b an d tw o -m o d e sq ueezed s ta te w ith th e sq ueez in g b an dw id th g 0 .

n u m b ers : g 0 20k H z, a 3 m , v 2 /m 9cm /s

o u tp u t f lu x o f ap p ro x . 680 a to m s /m s

sq ueez in g r 0 2 (la rg e)

Case 2: sequential pairsCase 2: sequential pairs

• Situation analogous to parametric downconversion

• Setup:

• State vector in perturbation theory

with wave function consisting of four pieces

• After postselection "one atom left" and "one atom right"

| eff fLRx,y 1 x 1

y 1 x 1

ydxdy|vac

| 1, 1LR | 1, 1LR

F 1| 1

|0| 1

symmetric potential

collisions

| t fx,y, t 1 x 1

ydx dy |vac

fx,y fLRx, y fRLx, y fLLx,y fRRx,y

6. Conclusions6. Conclusions

• Entangled states may be useful in precission measurements.

• Spin squeezed states can be generated with current technology.

- Collisions between atoms build up the entanglement.- One can achieve strongly spin squeezed states.

• The generation can be accelerated by using short pulses.

• The entanglement is very robust.

• Atoms can be outcoupled: squeezed atomic beams.

Quantum repeaters with atomic ensemblesQuantum repeaters with atomic ensembles

SFB Coherent Control€U TMR

€U EQUIP (IST)

L. M. Duan

M. Lukin

P. Zoller J.I.C.

(Nature, November 2001)

Quantum communication:Quantum communication:

Classical communication: Quantum communication:

Quantum Mechanics provides a secure way of secret communication

AliceBob Alice

Bob

Classical communication:

AliceBob

Quantum communication:

AliceBob

Eve

0

1010 1

1

jÁi jÁi

jÁi

jÁi jÁi

0

1010 1

1

½jÁi jÁi

jÁi½

Eve

Problem: decoherence.

We cannot know whether this is due to decoherence or to an eavesdropper.

Probability a photon arrives:

2. States are distorted:

Alice Bob

1. Photons are absorbed:

Quantum communication is limitedto short distances (< 50 Km).

j ª i ½

P =e_ L=L 0

In practice: photons.

laser

optical fiberphotons

vertical polarization

horizontal polarization

j0i = ay0jvaci

j1i = ay1jvacijÁi

laser repeater

Questions:

1. Number of repetitions

2. High fidelity:

3. Secure against eavesdropping.

j ª i j ª i½

< eL =L 0

F = hª j½jª i ' 1

Solution: Quantum repeaters.(Briegel et al, 1998).

OutlineOutline

1. Quantum repeaters:

2. Implementations:

1. With trapped ions.

2. With atomic ensembles.

3. Conclusions

1. Quantum repeaters1. Quantum repeaters

The goal is to establish entangled pairs:

(i) Over long distances.

(ii) With high fidelity.

(iii) With a small number of trials.

Once one has entangled states, one can use the Ekert protocol for secret communication.(Ekert, 1991)

Establish pairs over a short distance Small number of trials

Connect repeaters

Correct imperfections

Long distance

High fidelity

Key ideas:Key ideas:

1. Entanglement creation:

2. Connection:

3. Pufication:

4. Quantum communication:

2. Implementation with trapped ions2. Implementation with trapped ions

ion A ion Blaser

laser

ion A

ion B

Internal states

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

- If no detection, pump back and start again.

- If detection, an entangled state is created.

Entanglement creation:Entanglement creation:

j0i j0ij1i j1i

(Cabrillo et al, 1998)

jxi jxi

Initial state:

After laser pulse:

Evolution:

Detection:

Description:Description:

j0; 0i jvaci

j0; 0i jvaci + ²(bk j0; 1i j1k i + ak j1; 0i j1k i ) + o(²2)

bk j0; 1i § ak j1; 0i ' j0; 1i § j1; 0i

ion A ion B

j0i j0ij1i j1i

jxi jxi

(j0i + ²jxi )A (j0i + ²jxi )B jvaci£j0; 0i + ²j0; xi + ²jx; 0i + o(x2)

¤jvaci

Repeater:Repeater:

Entanglementcreation

Entanglementcreation

Gate operations:ConnectionPurification

3 Implementation with atomic ensembles3 Implementation with atomic ensembles

Internal states

- Weak (short) laser pulse, so that few atoms are excited.

- If no detection, pump back and start again.

- If detection, an entangled state is created.

j0ij1i

Atomic cell

Atomic cell

jxi

Initial state:

After laser pulse:

Evolution:

Detection:

j0i ­ n j0i ­ n jvaci

j0i ­ n j0i ­ n jvaci

+ photons in several directions (but not towards the detectors)

+ 2 photon towards the detectors and others in several directions

+ 1 photon towards the detectors and others in several directions

1 photon towards the detectors and others in several directions

+ 2 photon towards the detectors and others in several directions

Description:Description:

negligible

do not spoil the entanglement

(j0i + ²jxi )­ n (j0i + ²jxi ) ­ n jvaci

ayj =1pn

nX

k=1

ei 2¼kj =n j1i A n h0j

ay0 =1pn

nX

k=1

j1iA nh0j

Atomic „collective“ operators:Atomic „collective“ operators:

and similarly for b

Entanglement creation:

Measurement:

Sample A

Sample B

Apply operator

Apply operator:

(ay § by)

a

Photons emitted in the forward direction are the ones that excite this atomic „mode“.Photons emitted in other directions excite other (independent) atomic „modes“.

(A) Ideal scenareo(A) Ideal scenareo

After click:

(1)

(2)

After click:

Thus, we have the state:

Sample A

Sample R

Sample B

A.1 Entanglement generation:

(ay+ r y)j0; 0i

(by+ ~r y)j0; 0i

(by+ ~r y)(ay + r y) j0; 0i

A.2 Connection:

If we detect a click, we must apply the operator:

Otherwise, we discard it.

We obtain the state:

(r + ~r )

(by + ay)j0; 0i

(by+ ~r y)(ay + r y) j0; 0i

jr ij~r i

A.3 Secret Communication:

- Check that we have an entangled state:

One can use this method to send information.

• Enconding a phase:

• Measurement in A

• Measurement in B:

(~by + ~ay)(by + ay)j0; 0i

(~by + ei±~ay)(by+ ay)j0; 0i

(a + ~a)

(b+ ~b)

The probability of different outcomes +/- depends on ±

(B) Imperfections:(B) Imperfections:

- Spontaneous emission in other modes:

No effect, since they are not measured.

- Detector efficiency, photon absorption in the fiber, etc:

More repetitions.

- Dark counts:

More repetitions

- Systematic phaseshifts, etc:

Are directly purified

(C) Efficiency:(C) Efficiency:

Fix the final fidelity: F

Number of repetitions: rN log2 N

Example:

Detector efficiency: 50%

Length L=100 L0

Time T=10 T06

(to be compared with T=10 T0 for direct communication)43

Advantages of atomic ensembles:Advantages of atomic ensembles:

1. No need for trapping, cooling, high-Q cavities, etc.

2. More efficient than with single ions: the photons that change the collective mode go in the forward direction (this requires a high optical thickness).

Photons connected to the collective mode.

Photons connected to other modes.

4. Purification is built in.

3. Connection is built in. No need for gates.

4. Conclusions4. Conclusions

• Quantum repeaters allow to extend quantum communication over long distances.

• They can be implemented with trapped ions or atomic ensembles.

• The method proposed here is efficient and not too demanding:

1. No trapping/cooling is required.

2. No (high-Q) cavity is required.

3. Atomic collective effects make it more efficient.

4. No high efficiency detectors are required.

Institute for Theoretical PhysicsInstitute for Theoretical Physics

€

FWF SFB F015:„Control and Measurement of Coherent Quantum Systems“

EU networks:„Coherent Matter Waves“, „Quantum Information“

EU (IST):„EQUIP“

Austrian Industry:Institute for Quantum Information Ges.m.b.H.

P. ZollerJ. I. Cirac

Postdocs: - L.M. Duan (*) - P. Fedichev - D. Jaksch - C. Menotti (*) - B. Paredes - G. Vidal - T. Calarco

Ph D: - W. Dur (*) - G. Giedke (*) - B. Kraus - K. Schulze