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)
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`)
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`}=;
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