Preventing Disentanglement by

Symmetry Manipulations

G. Gordon, A. Kofman, G. Kurizki

Weizmann Institute of Science, Rehovot 76100, Israel

Sponsors: EU, ISF

Outline

• Decoherence mechanisms• General formalism• Modulation schemes• Numerical example• Conclusions

Decoherence Scenarios: Single Particle

Ion trapCold atom in (imperfect) optical lattice

Ion in cavity

Kreuter et al. PRL 92 203002 (2004)

Keller et al. Nature 431, 1075 (2004)

Häffner et al. Nature 438 643 (2005)

Jaksch et al. PRL 82, 1975 (1999)Mandel et al. Nature 425, 937 (2003)

Single Particle Solution

t)=e-J(t)(0)

J(t)=2s-11 d G()Ft(+a)

Reservoir coupling spectrum

Spectral intensity of modulation

G()!()|()|2

Ft()=|t()|2

F(t)=|h(0)|(t)i|2=e-2< J(t)

Fidelity:

A. G. Kofman and G. Kurizki, Nature 405, 546 (2000), PRL 87, 270405 (2001), PRL 93,130406(2004)

Impulsive phase modulation(Caused by Repetitive Weak Pulses)t)=ei[t/]¿J=2 G(a+/)

G()

Ft()

Dynamic decoupling.Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)

t)i=kk(t)|ki|gi+(t)|vaci|ei

Decoherence Scenarios: Many Particles

Ions’ vibrations in trap

Ions in cavity

“Sudden Death”, Yu & Eberly PRL 93 140404 (2004)

Coupled atoms’ vibrations in imperfect optical lattice

Lisi & Mølmer, PRA 66, 052303 (2002); Sherson & Mølmer, PRA 71, 033813 (2005)

|2i|1i

|gi

|2i|1i

|gi

(a)(b)

• Particles:– Ions– Cold atoms

• Bath:– Cavity modes – Vibrational modes

• Bath-particle coupling• Modulations:

– AC Stark Shifts: RF fields, Lasers– Coupling modulation: On-off switch

The System

a,1(t)b,2(t)

k,a,1

k,b,1

|ki

The Multipartite Wavefunction

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|2i|1i

|gi

|(t)i = kk(t)|ki |gia|gib +

a,1(t)|vaci |1ia|gib +

a,2(t)|vaci |2ia|gib +

b,1

(t)|vaci |gia|1ib +

b,2

(t)|vaci |gia|2ib

a b

t)=e-J(t)(0)

Jjj',nn'(t) = s0tdt's0

t'dt''jj',nn'(t'-t'')Kjj',nn'(t',t'')eij,nt'-ij',n't''

jj',nn'(t)=s d Gjj',nn'()e-i t

Gjj',nn‘~-2kk,j,n*k,j',n'k)

Bath Matrix Modulation MatrixKjj',nn'(t,t')=*

j,n(t)j',n'(t')

j,n(t)=eis0 d j,n()

Decoherence Matrix

0

0

0

0

2,

1,

2,

1,

2,

1,

2,

1,

22,21,22,21,

12,11,12,11,

22,21,22,21,

12,11,12,11,

b

b

a

a

JJJJ

JJJJ

JJJJ

JJJJ

b

b

a

a

bbbbbaba

bbbbbaba

ababaaaa

ababaaaa

e

t

t

t

t

The Multipartite SolutionGordon, Kurizki, Kofman J. Opt. B. 7 283, (2005); Opt. Comm. (in press)

Decoherence Matrix Elements

0

0

0

0

2,

1,

2,

1,

2,

1,

2,

1,

22,21,22,21,

12,11,12,11,

22,21,22,21,

12,11,12,11,

b

b

a

a

JJJJ

JJJJ

JJJJ

JJJJ

b

b

a

a

bbbbbaba

bbbbbaba

ababaaaa

ababaaaa

e

t

t

t

t

Diagonal elements: Individual particle decoherence

Jjj,nn(t)=2s-11 d Gjj,nn()Ft,j,n(+j,n)

Off-diagonal elements: Cross-decoherence

Jjj',nn'(t) = s0tdt's0

t'dt''jj',nn'(t'-t'')*j,n(t`)j`n`(t``)eij,nt'-ij',n't''

F(t)=|h(0)|(t)i|2Definitions:

Mixing parameters: cj,n(t)=j,n(t)/1,1(t)

Decay parameter: A(t)=1,1(t)(j,n|cj,n(t)|2)1/2

F(t)=Fp(t)Fe(t)

Fp(t)=|A(t)|2

Fe(t)=|1,1(0)|2 |j=12n=1

2c*j,n(0)cj,n(t)|2

j=12n=1

2|cj,n(t)|2

Population Preservation

Entanglement Preservation

The Fidelity

F(t) of single particle

Population preservation: probability of having a particle in an excited state

|(0)i=1/√2(|gia|1ib+|1ia|gib)

a(t)=1/√2 e-Ja(t)

b(t)=1/√2 e-Jb(t)

Fp(t)=(e-2Ja(t)+e-2Jb(t))/2

Fe(t)=1/2+e- J(t)/(1+e-2 J(t))J(t) = Jb(t)-Ja(t)

No Cross-decoherence, different decoherence rates:

Initial entangled state:

Entanglement preservation:

Given that a particle is in an excited state, a measure of entanglement preservation compared to initial state.

The Fidelity: Example

Modulation Schemes Tasks

No Modulation

N different independent particlesN identical independent particles

Decoherence Free SubspaceN decoherence free qubits

Viola et al. PRL 85, 3520 (2000); Wu & Lidar, PRL 88, 207902 (2002)

,Global Modulation

• Two three-level particles

• Coupling: Gaussian, Gjj`,nn`()/ exp(-2/j,n2)exp(-2/j`,n`

2)– Different for each particle– Cross-decoherence

• Impulsive phase modulation j,n(t)=ei[t/j,n]j,n

– Global Scheme: Identical modulation to all particles

– Local Scheme: Addressability, specific modulations

• Initial Entangled State: |(0)i=1/√2(|-ia|gib+|gia|-ib)|-ij=1/√2(|1ij-|2ij) ”dark state”

Numerical Example: Setup

J(t)

Global Modulation

• General Decoherence Matrix

• Cross-coupling particles,Different coupling to bath

• Population lossEntanglement loss

Condition:

j,n(t)=(t) 8 j,n

Jjj`,nn`(t)=2s-11 d Gjj`,nn`()Ft(+j,n) 0

Decoherence MatrixDecoherence Matrix Elements

FidelityFFp

Fe

Numerical Example: Global Modulation No Symmetry

• Diagonal Decoherence Matrix

• Effectively: N different independent particles

• Separated particles,Coupled to different baths

• Population lossEntanglement loss

J(t)Task 1: Eliminating cross-decoherence

Jjj',nn'(t) = s0tdt's0

t'dt''jj',nn'(t'-t'')*j,n(t`)j`n`(t``)eij,nt'-ij',n't''=0 8 j j`, n n`

Condition:

j,n(t)j`,n`(t) 8 j,j`,n,n`

Fidelity

Decoherence Matrix ElementsDecoherence Matrix

FFp

Fe

Numerical Example: Local modulations Eliminate cross-decoherence

• Decoherence Matrix / Identity Matrix

• Imposes permutation symmetry

• Effectively: N independent identical particles• Separated particles, identical coupling to baths

• Reduce problem to single particle decoherence control

• Population lossEntanglement preservation

J(t)

Task 2: Equating decoherence rates

Decaying Entangled state|(t)i=e-J(t)|(0)i

Condition:

j,n(t)j`,n`(t) 8 j,j`,n,n`

Jjj,nn(t)=2s-11 d Gjj,nn()Ft,j,n(+j,n) =J(t)

G()

Ft()

G()

Ft()=

Fidelity

Decoherence Matrix ElementsDecoherence Matrix

FFp

Fe

Numerical Example: Equating decoherence rates

J(t)

Task 3: Equating decoherence and cross-decoherence

• All Decoherence Matrix Elements Equal

• Imposes permutation symmetry

• Cross-coupled particles,identical coupling to the same bath

• Anti-symmetric state = Decoherence-Free Subspace

Condition:

j,n(t)¼j`,n`(t) 8 j,j`,n,n`

Jjj`,nn`(t)=J(t)Very difficult

J(t)• For N three-level particles• Equating intraparticle decoherence

and cross-decoherence of each particle• Eliminating interparticle cross-decoherence

• Anti-symmetric state of each particle |-ij=1/√2(|1ij-|2ij)

= Decoherence Free Subspace

N decoherence free qubits

Optimal modulation scheme

Condition:

j,n(t)¼j`,n`(t) 8 j=j`,n,n`

j,n(t)j`,n`(t) 8 j j`,n,n’

Fidelity

Decoherence Matrix ElementsDecoherence Matrix

FFp

Fe

Numerical Example: Optimal scheme

|(0)i=1/√2(|-ia|gib+|gia|-ib)

|-ij=1/√2(|1i2-|2ij)

Numerical Example: Summary

Suggested Experimental Setup:Multiple 40Ca+ Ions in Cavity

Experimental parameters:Finesse ¼ 3500032D5/2 a = 729 nm Cavity mode width ¼ 12

GHz

Single particle:No modulation: Lifetime = 1.168 sWith modulation: Lifetime ¼ 1.4 s

Required impulsive phase modulation rate ~

Multiple ions in cavity:Position in cavity: a-b ¼ 15%Three level system: |gi = 42S1/2

|1i = 32D3/2

|2i = 32D5/2

1/2 = 1.026

Kreuter et al. PRL 92 203002 (2004);Barton et al. PRA 62 032503 (2000)

Single ion in cavity:

Two ions in each cavity + Local modulations1/√2(|gia|1ib-|1ia|gib) = DFS

• Local modulations can– Impose permutation symmetry– Introduce a Decoherence-Free Subspace– Reduce the task of multipartite disentanglement

to that of a single relaxing particle

• Universal dynamical decoherence control formalism gives the modulations’ conditions for each task

• Optimal modulation scheme for N three-level particles– Can impose many-particle DFS

Conclusions

Thank You !!!

Modulation Criteria

Global modulation

Elimination of cross-decoherence

Creation of DFS

G()

Ft,j,n()=Ft,j`,n`()

G()

Ft,j,n()

G()

Ft,j,n()

Ft,j`,n`()

Ft,j`,n`()

j,n(t)=j`,n`(t)

j,n(t)j`,n`(t)

j,n(t)¼j`,n`(t)

Multilevel Cross-Decoherence

No modulation

Global modulation

Creation of DFS

G()

Ft,j,n(+j)

G()

j,n(t)=

j,n(t)j`,n`(t)

j,n(t)¼j`,n`(t)

Ft,j`,n(+j`)

G()

Ft,j,n(+j) Ft,j`,n(+j`)

Ft,j,n(+j) Ft,j`,n(+j`)

Quasi-periodic Modulation

j,n(t)=l l e-ilt

G()

Ft,j,n()

l l+1 l+2 l+3l-1l-2l-3

J(t)=2l l G(+l)

General Bath Formalism

Jjj',nn'(t) = s0tdt's0

t'dt''jj',nn'(t'-t'')Kjj',nn'(t',t'')eij,nt'-ij',n't''

jj',nn'(t)=s d Gjj',nn'()e-i t

Gjj',nn‘~-2kk,j,n*k,j',n'k)

Bath Matrix

Decoherence Matrix

The Same Bath:

k,j,n=k 8 j,n

Separate Baths: (independent particles)k,j,nk,j`,n`=0 8 jj`,n,n`,k

Particle j coupled to modes {kj}Particle j` coupled to modes {kj`} {kj}Å{kj`}=;

"Dark State”

|0i

|1i

|2i

1

V(t)

2*1

*2

1

0

0

0

0)"("

V

V

V

VtJ

No Cross-Decoherence, levels coupled to separate baths

|0i

|1i

|2i

1

12

2*12

121)(

tJ

11

11)()"(" tjtJ

Exploiting Cross-Decoherence to create “dark state”